Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc Tables

Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­

Chapter 2  Organizing and Summarizing Data • Complement Rule • Factorial Chapter 10  Hypothesis Tests Regarding a Parameter Chapter 14  Inference on the Least-Squares Regression Model and Multiple Regression
frequency • Class midpoint: The sum of consecutive lower class limits
c
P1E 2 = 1 - P1E2 n! = n # 1n - 12 # 1n - 22 # g # 3 # 2 # 1 Test Statistics • Standard Error of the Estimate • Confidence Interval about the Mean Response of y, yn
g 1yi - yn i 2 g residuals
• Relative frequency = divided by 2.
sum of all frequencies • Multiplication Rule for Independent Events pn - p0
2 2
1x* - x2 2
C n g 1xi - x2 2
yn { ta>2 # se
• Permutation of n objects taken r at a time:  x - m0 1
• z0 = • t0 = se = = +
P1E and F 2 = P1E2 # P1F 2 p0 11 - p0 2 s  1n C n- 2 C n- 2
Chapter 3  Numerically Summarizing Data

n!
gxi gxi fi
n Pr = C n • Standard error of b1
• Multiplication Rule for n Independent Events 1n - r2! 1n - 12s2 where x* is the given value of the explanatory variable and

gfi
• x20 =
P1E and F and G g 2 = P1E2 # P1F2 # P1G2 # g
• Population Mean:  m = • Population Mean from Grouped Data:  m = s20 se ta>2 is the critical value with n - 2 degrees of freedom.
N • Combination of n objects taken r at a time: sb1 =

gxi gxi fi
Chapter 11  Inferences on Two Samples 2g 1xi - x2 2 • Prediction Interval about an Individual Response, yn
• Conditional Probability Rule
gfi
n!
• Sample Mean:  x = • Sample Mean from Grouped Data:  x = nC r = • Test Statistic for the Slope of the Least-Squares Regression Line 1x* - x2 2
g 1xi - x2 2
yn { ta>2 # se
n r!1n - r2! • Test Statistic Comparing Two Population Proportions • Test Statistic Comparing Two Means (Independent Sampling) 1
P1E and F 2 N1E and F 2 1+ +
gwi xi
P1F  E2 = (Independent Samples) C

= b1 - b1 b1 - b1 n
• Range = Largest Data Value - Smallest Data Value P1E2 N1E2 1x1 - x2 2 - 1m1 - m2 2 t0 = =
gwi
• Permutations with Repetition: sb1
• Weighted Mean:  xw = pn 1 - pn 2 - 1p1 - p2 2 x1 + x2 t0 =
se n 2g 1xi - x2 2  here x* is the given value of the explanatory variable and
w

• Population Standard Deviation: • General Multiplication Rule n! z0 = where

pn = s21 s22

1 gxi 2 2
n1 + n2 + ta>2 is the critical value with n - 2 degrees of freedom.
n1! # n2! # g # nk!
1 1
P1E and F2 = P1E2 # P1F  E2
gx 2i
• Population Standard Deviation from Grouped Data: 2pn 11 - pn 2 + C n1 n2 • Confidence Interval for the Slope of the Regression Line

g 1xi - m2 2 1 gxi fi 2 2
B n1 n2
gx 2i fi -
- se
gfi
b1 { ta>2 #
g 1xi - m2 fi
N Chapter 6  Discrete Probability Distributions • Confidence Interval for the Difference of Two Means
s = = 2 • Confidence Interval for the Difference of Two Proportions (Independent Samples)
gfi gfi
C N S N 2g 1xi - x2 2
s = = • Mean (Expected Value) of a Discrete Random Variable • Mean and Standard Deviation of a Binomial Random Variable (Independent Samples)

mX = gx # P1x2
B R s21 s22
• Sample Standard Deviation 1x1 - x2 2 { ta>2 where ta>2 is computed with n - 2 degrees of freedom.
1 gxi 2
pn 1 11 - pn 1 2 pn 2 11 - pn 2 2 +
mX = np sX = 2np11 - p2 C n1
gx 2i -
• Sample Standard Deviation from Grouped Data: 1pn 1 - pn 2 2 { za>2 + n2

1 gxi fi 2 2
2

g 1xi - x 2
C n1 n2 Chapter 15  Nonparametric Statistics
gx 2i fi
• Standard Deviation of a Discrete Random Variable • Poisson Probability Distribution Function
gfi
Note: ta>2 is found using the smaller of n1 - 1 or n2 - 1
g 1xi - m2 2fi
2

sX = 3 g 1x - m2 2 # P1x2 = 3 g 3x 2P1x24 - m2X

n
s = = - • Test Statistic for Matched-Pairs Data degrees of freedom. • Test Statistic for a Runs Test for Randomness Large-Sample Case 1 n + 302
B 1 gfi 2 - 1 gfi - 1
C n-1 S n- 1 1lt2 x
s = = P1x2 = e -lt x = 0, 1, 2, c Small-Sample Case  If n1 … 20 and n2 … 20, the test statistic in
R x! d - md • Test Statistic for Comparing Two Population Standard n1n + 12
• Population Variance:  s2 • Binomial Probability Distribution Function t0 = the runs test for randomness is r, the number of runs. T -
sd  1n Deviations 4
x- m • Mean and Standard Deviation of a Poisson Random Variable Large-Sample Case  If n1 7 20 or n2 7 20, the test statistic is z0 =
• Sample Variance:  s2 • Population z-score:  z = P1x2 = nCx px 11 - p2 n - x
s where d is the mean and sd is the standard deviation of the s21 r - mr n1n + 12 12n + 12
• Empirical Rule: If the shape of the distribution is bell- mX = lt sX = 2lt F0 = z0 = where
differenced data. s22 sr C 24
shaped, then x- x Chapter 7  The Normal Distribution
• Sample z-score:  z = where T is the test statistic from the small-sample case.
• Approximately 68% of the data lie within 1 standard s • Confidence Interval for Matched-Pairs Data • Finding a Critical F for the Left Tail 2n1n2 2n1n2 12n1n2 - n2
• Standardizing a Normal Random Variable • Finding the Score:  x = m + zs mr = + 1 and sr =
deviation of the mean sd n B n2 1n - 12 • Test Statistic for the Mann–Whitney Test
d { ta>2 #
• Interquartile Range:  IQR = Q3 - Q1 1
• Approximately 95% of the data lie within 2 standard x- m F1 - a,n1 - 1,n2 - 1 = Small-Sample Case 1 n1 " 20 and n2 " 202
z = 1n Fa,n2 - 1,n1 - 1 • Test Statistic for a One-Sample Sign Test
deviations of the mean Lower fence = Q1 - 1.51IQR2 If S is the sum of the ranks corresponding to the sample from
• Lower and Upper Fences:  s
Small-Sample Case 1n " 252
• Approximately 99.7% of the data lie within 3 standard Upper fence = Q3 + 1.51IQR2 Note: ta>2 is found using n - 1 degrees of freedom. population X, then the test statistic, T, is given by
deviations of the mean Chapter 8  Sampling Distributions Two-Tailed Left-Tailed Right-Tailed
• Five-Number Summary n1 1n1 + 12
Chapter 12  Inference on Categorical Data T = S-
• Mean and Standard Deviation of the Sampling Distribution • Mean and Standard Deviation of the Sampling Distribution H0 : M = M0 H0 : M = M0 H0 : M = M0 2
Minimum, Q1, M, Q3, Maximum
of x of pn • Expected Counts (when testing for goodness of fit) • Chi-Square Test Statistic H1 : M ≠ M0 H1 : M 6 M0 H1 : M 7 M0 Note: The value of S is always obtained by summing the ranks of

x20 = a = a
Chapter 4  Describing the Relation between Two Variables s p11 - p2 Ei = mi = npi for i = 1, 2, c, k 1observed - expected2 2 1Oi - Ei 2 2 The test statistic, k, is the The test statistic, The test statistic, the sample data that correspond to MX in the hypothesis.
mx = m and sx = m pn = p and s pn = smaller of the number of k, is the number k, is the number Large-Sample Case 1 n1 + 202 or 1 n2 + 202
B n
aa s ba s b
xi - x yi - y • Residual = observed y - predicted y = y - yn 2n • Expected Frequencies (when testing for independence or expected Ei
minus signs or plus signs. of plus signs. of minus signs.
homogeneity of proportions) i = 1, 2, c, k n1n2
x y x T -
• Correlation Coefficient:  r = • R2 = r 2 for the least-squares regression model • Sample Proportion:  pn = Large-Sample Case 1n + 252 The test statistic, z0, is 2
n-1 yn = b1x + b0 n 1row total21column total2 All Ei Ú 1 and no more than 20% less than 5. z0 =
Expected frequency = n n1n2 1n1 + n2 + 12
• The equation of the least-squares regression line is 2 Chapter 9  Estimating the Value of a Parameter table total • Test Statistic for Comparing Two Proportions 1k + 0.52 -
• The coefficient of determination, R , measures the 2 B 12
proportion of total variation in the response variable that is (Dependent Samples) z0 =
sy
yn = b1x + b0, where yn is the predicted value, b1 = r # Confidence Intervals 1n • Test Statistic for Spearman’s Rank Correlation Test
explained by the least-squares regression line.
• A 11 - a2 # 100% confidence interval about p is
sx Sample Size 1f12 - f21 2 2 2
is the slope, and b0 = y - b1x is the intercept. • To estimate the population proportion with a margin of x20 = 6gd 2i
pn 11 - pn 2 error E at a 11 - a2 # 100% level of confidence: f12 + f21 where n is the number of minus and plus signs and k is obtained rs = 1 -
pn { za>2 # as described in the small sample case. n1n2 - 12
za>2 2 Chapter 13  Comparing Three or More Means
Chapter 5  Probability B n n = pn 11 - pn 2 a b rounded up to the next integer,
E • Test Statistic for the Wilcoxon Matched-Pairs where di = the difference in the ranks of the two
• Empirical Probability • Addition Rule for Disjoint Events • A 11 - a2 # 100% confidence interval about m is where pn is a prior estimate of the population proportion,
• Test Statistic for One-Way ANOVA • Test Statistic for Tukey’s Test after One-Way ANOVA
Signed-Ranks Test observations in the i th ordered pair.
za>2 2 Mean square due to treatment 1x2 - x1 2 - 1m2 - m1 2 x2 - x1 • Test Statistic for the Kruskal–Wallis Test
x { ta>2 #
s MST Small-Sample Case 1n " 302
frequency of E P1E or F 2 = P1E2 + P1F 2 or n = 0.25 a b rounded up to the next integer when no F = = q = =

a
E

P1E2 ≈ 1n Mean square due to error MSE

s # 12
1 s # 12
1 Two-Tailed Left-Tailed Right-Tailed 12 1 ni 1N + 12 2
number of trials of experiment prior estimate of p is available. a + b a + b H = c Ri - d
• Addition Rule for n Disjoint Events B2 n1 n2 B2 n1 n2 N1N + 12 ni 2
Note: ta>2 is computed using n - 1 degrees of freedom. where H0: MD = 0 H0: MD = 0 H0: MD = 0
• Classical Probability • To estimate the population mean with a margin of error E R21 R22 R2k
P1E or F or G or g 2 = P1E2 + P1F 2 + P1G2 + g
za>2 # s 2
12
• A 11 - a2 # 100% confidence interval about s is
2 2 2

n1 1x1 - x2 + n2 1x2 - x2 + g + nk 1xk - x2 H1: MD ≠ 0 H1: MD 6 0 H0: MD 7 0 = J + + g+ R - 31N + 12

number of ways that E can occur N1E2
• General Addition Rule at a 11 - a2 # 100% level of confidence: n = a b MST =
k- 1 Test Statistic: T is the Test Statistic: Test Statistic:
N1N + 12 n1 n2 nk
P1E2 = = E
number of possible outcomes N1S2 1n - 12s2 1n - 12s2 smaller of T + or T - T = T+ T = T- 
P1E or F 2 = P1E2 + P1F 2 - P1E and F 2 6s6 rounded up to the next integer. 1n1 - 12s21 + 1n2 - 12s22 + g + 1nk - 12s2k where Ri is the sum of the ranks in the ith sample.
B x2a>2 B x21 - a>2 MSE =

n- k

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 Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­

Chapter 2  Organizing and Summarizing Data • Complement Rule • Factorial Chapter 10  Hypothesis Tests Regarding a Parameter Chapter 14  Inference on the Least-Squares Regression Model and Multiple Regression
frequency • Class midpoint: The sum of consecutive lower class limits
c
P1E 2 = 1 - P1E2 n! = n # 1n - 12 # 1n - 22 # g # 3 # 2 # 1 Test Statistics • Standard Error of the Estimate • Confidence Interval about the Mean Response of y, yn
g 1yi - yn i 2 g residuals
• Relative frequency = divided by 2.
sum of all frequencies • Multiplication Rule for Independent Events pn - p0
2 2
1x* - x2 2
C n g 1xi - x2 2
yn { ta>2 # se
• Permutation of n objects taken r at a time:  x - m0 1
• z0 = • t0 = se = = +
P1E and F 2 = P1E2 # P1F 2 p0 11 - p0 2 s  1n C n- 2 C n- 2
Chapter 3  Numerically Summarizing Data

n!
gxi gxi fi
n Pr = C n • Standard error of b1
• Multiplication Rule for n Independent Events 1n - r2! 1n - 12s2 where x* is the given value of the explanatory variable and

gfi
• x20 =
P1E and F and G g 2 = P1E2 # P1F2 # P1G2 # g
• Population Mean:  m = • Population Mean from Grouped Data:  m = s20 se ta>2 is the critical value with n - 2 degrees of freedom.
N • Combination of n objects taken r at a time: sb1 =

gxi gxi fi
Chapter 11  Inferences on Two Samples 2g 1xi - x2 2 • Prediction Interval about an Individual Response, yn
• Conditional Probability Rule
gfi
n!
• Sample Mean:  x = • Sample Mean from Grouped Data:  x = nC r = • Test Statistic for the Slope of the Least-Squares Regression Line 1x* - x2 2
g 1xi - x2 2
yn { ta>2 # se
n r!1n - r2! • Test Statistic Comparing Two Population Proportions • Test Statistic Comparing Two Means (Independent Sampling) 1
P1E and F 2 N1E and F 2 1+ +
gwi xi
P1F  E2 = (Independent Samples) C

= b1 - b1 b1 - b1 n
• Range = Largest Data Value - Smallest Data Value P1E2 N1E2 1x1 - x2 2 - 1m1 - m2 2 t0 = =
gwi
• Permutations with Repetition: sb1
• Weighted Mean:  xw = pn 1 - pn 2 - 1p1 - p2 2 x1 + x2 t0 =
se n 2g 1xi - x2 2  here x* is the given value of the explanatory variable and
w

• Population Standard Deviation: • General Multiplication Rule n! z0 = where

pn = s21 s22

1 gxi 2 2
n1 + n2 + ta>2 is the critical value with n - 2 degrees of freedom.
n1! # n2! # g # nk!
1 1
P1E and F2 = P1E2 # P1F  E2
gx 2i
• Population Standard Deviation from Grouped Data: 2pn 11 - pn 2 + C n1 n2 • Confidence Interval for the Slope of the Regression Line

g 1xi - m2 2 1 gxi fi 2 2
B n1 n2
gx 2i fi -
- se
gfi
b1 { ta>2 #
g 1xi - m2 fi
N Chapter 6  Discrete Probability Distributions • Confidence Interval for the Difference of Two Means
s = = 2 • Confidence Interval for the Difference of Two Proportions (Independent Samples)
gfi gfi
C N S N 2g 1xi - x2 2
s = = • Mean (Expected Value) of a Discrete Random Variable • Mean and Standard Deviation of a Binomial Random Variable (Independent Samples)

mX = gx # P1x2
B R s21 s22
• Sample Standard Deviation 1x1 - x2 2 { ta>2 where ta>2 is computed with n - 2 degrees of freedom.
1 gxi 2
pn 1 11 - pn 1 2 pn 2 11 - pn 2 2 +
mX = np sX = 2np11 - p2 C n1
gx 2i -
• Sample Standard Deviation from Grouped Data: 1pn 1 - pn 2 2 { za>2 + n2

1 gxi fi 2 2
2

g 1xi - x 2
C n1 n2 Chapter 15  Nonparametric Statistics
gx 2i fi
• Standard Deviation of a Discrete Random Variable • Poisson Probability Distribution Function
gfi
Note: ta>2 is found using the smaller of n1 - 1 or n2 - 1
g 1xi - m2 2fi
2

sX = 3 g 1x - m2 2 # P1x2 = 3 g 3x 2P1x24 - m2X

n
s = = - • Test Statistic for Matched-Pairs Data degrees of freedom. • Test Statistic for a Runs Test for Randomness Large-Sample Case 1 n + 302
B 1 gfi 2 - 1 gfi - 1
C n-1 S n- 1 1lt2 x
s = = P1x2 = e -lt x = 0, 1, 2, c Small-Sample Case  If n1 … 20 and n2 … 20, the test statistic in
R x! d - md • Test Statistic for Comparing Two Population Standard n1n + 12
• Population Variance:  s2 • Binomial Probability Distribution Function t0 = the runs test for randomness is r, the number of runs. T -
sd  1n Deviations 4
x- m • Mean and Standard Deviation of a Poisson Random Variable Large-Sample Case  If n1 7 20 or n2 7 20, the test statistic is z0 =
• Sample Variance:  s2 • Population z-score:  z = P1x2 = nCx px 11 - p2 n - x
s where d is the mean and sd is the standard deviation of the s21 r - mr n1n + 12 12n + 12
• Empirical Rule: If the shape of the distribution is bell- mX = lt sX = 2lt F0 = z0 = where
differenced data. s22 sr C 24
shaped, then x- x Chapter 7  The Normal Distribution
• Sample z-score:  z = where T is the test statistic from the small-sample case.
• Approximately 68% of the data lie within 1 standard s • Confidence Interval for Matched-Pairs Data • Finding a Critical F for the Left Tail 2n1n2 2n1n2 12n1n2 - n2
• Standardizing a Normal Random Variable • Finding the Score:  x = m + zs mr = + 1 and sr =
deviation of the mean sd n B n2 1n - 12 • Test Statistic for the Mann–Whitney Test
d { ta>2 #
• Interquartile Range:  IQR = Q3 - Q1 1
• Approximately 95% of the data lie within 2 standard x- m F1 - a,n1 - 1,n2 - 1 = Small-Sample Case 1 n1 " 20 and n2 " 202
z = 1n Fa,n2 - 1,n1 - 1 • Test Statistic for a One-Sample Sign Test
deviations of the mean Lower fence = Q1 - 1.51IQR2 If S is the sum of the ranks corresponding to the sample from
• Lower and Upper Fences:  s
Small-Sample Case 1n " 252
• Approximately 99.7% of the data lie within 3 standard Upper fence = Q3 + 1.51IQR2 Note: ta>2 is found using n - 1 degrees of freedom. population X, then the test statistic, T, is given by
deviations of the mean Chapter 8  Sampling Distributions Two-Tailed Left-Tailed Right-Tailed
• Five-Number Summary n1 1n1 + 12
Chapter 12  Inference on Categorical Data T = S-
• Mean and Standard Deviation of the Sampling Distribution • Mean and Standard Deviation of the Sampling Distribution H0 : M = M0 H0 : M = M0 H0 : M = M0 2
Minimum, Q1, M, Q3, Maximum
of x of pn • Expected Counts (when testing for goodness of fit) • Chi-Square Test Statistic H1 : M ≠ M0 H1 : M 6 M0 H1 : M 7 M0 Note: The value of S is always obtained by summing the ranks of

x20 = a = a
Chapter 4  Describing the Relation between Two Variables s p11 - p2 Ei = mi = npi for i = 1, 2, c, k 1observed - expected2 2 1Oi - Ei 2 2 The test statistic, k, is the The test statistic, The test statistic, the sample data that correspond to MX in the hypothesis.
mx = m and sx = m pn = p and s pn = smaller of the number of k, is the number k, is the number Large-Sample Case 1 n1 + 202 or 1 n2 + 202
B n
aa s ba s b
xi - x yi - y • Residual = observed y - predicted y = y - yn 2n • Expected Frequencies (when testing for independence or expected Ei
minus signs or plus signs. of plus signs. of minus signs.
homogeneity of proportions) i = 1, 2, c, k n1n2
x y x T -
• Correlation Coefficient:  r = • R2 = r 2 for the least-squares regression model • Sample Proportion:  pn = Large-Sample Case 1n + 252 The test statistic, z0, is 2
n-1 yn = b1x + b0 n 1row total21column total2 All Ei Ú 1 and no more than 20% less than 5. z0 =
Expected frequency = n n1n2 1n1 + n2 + 12
• The equation of the least-squares regression line is 2 Chapter 9  Estimating the Value of a Parameter table total • Test Statistic for Comparing Two Proportions 1k + 0.52 -
• The coefficient of determination, R , measures the 2 B 12
proportion of total variation in the response variable that is (Dependent Samples) z0 =
sy
yn = b1x + b0, where yn is the predicted value, b1 = r # Confidence Intervals 1n • Test Statistic for Spearman’s Rank Correlation Test
explained by the least-squares regression line.
• A 11 - a2 # 100% confidence interval about p is
sx Sample Size 1f12 - f21 2 2 2
is the slope, and b0 = y - b1x is the intercept. • To estimate the population proportion with a margin of x20 = 6gd 2i
pn 11 - pn 2 error E at a 11 - a2 # 100% level of confidence: f12 + f21 where n is the number of minus and plus signs and k is obtained rs = 1 -
pn { za>2 # as described in the small sample case. n1n2 - 12
za>2 2 Chapter 13  Comparing Three or More Means
Chapter 5  Probability B n n = pn 11 - pn 2 a b rounded up to the next integer,
E • Test Statistic for the Wilcoxon Matched-Pairs where di = the difference in the ranks of the two
• Empirical Probability • Addition Rule for Disjoint Events • A 11 - a2 # 100% confidence interval about m is where pn is a prior estimate of the population proportion,
• Test Statistic for One-Way ANOVA • Test Statistic for Tukey’s Test after One-Way ANOVA
Signed-Ranks Test observations in the i th ordered pair.
za>2 2 Mean square due to treatment 1x2 - x1 2 - 1m2 - m1 2 x2 - x1 • Test Statistic for the Kruskal–Wallis Test
x { ta>2 #
s MST Small-Sample Case 1n " 302
frequency of E P1E or F 2 = P1E2 + P1F 2 or n = 0.25 a b rounded up to the next integer when no F = = q = =

a
E

P1E2 ≈ 1n Mean square due to error MSE

s # 12
1 s # 12
1 Two-Tailed Left-Tailed Right-Tailed 12 1 ni 1N + 12 2
number of trials of experiment prior estimate of p is available. a + b a + b H = c Ri - d
• Addition Rule for n Disjoint Events B2 n1 n2 B2 n1 n2 N1N + 12 ni 2
Note: ta>2 is computed using n - 1 degrees of freedom. where H0: MD = 0 H0: MD = 0 H0: MD = 0
• Classical Probability • To estimate the population mean with a margin of error E R21 R22 R2k
P1E or F or G or g 2 = P1E2 + P1F 2 + P1G2 + g
za>2 # s 2
12
• A 11 - a2 # 100% confidence interval about s is
2 2 2

n1 1x1 - x2 + n2 1x2 - x2 + g + nk 1xk - x2 H1: MD ≠ 0 H1: MD 6 0 H0: MD 7 0 = J + + g+ R - 31N + 12

number of ways that E can occur N1E2
• General Addition Rule at a 11 - a2 # 100% level of confidence: n = a b MST =
k- 1 Test Statistic: T is the Test Statistic: Test Statistic:
N1N + 12 n1 n2 nk
P1E2 = = E
number of possible outcomes N1S2 1n - 12s2 1n - 12s2 smaller of T + or T - T = T+ T = T- 
P1E or F 2 = P1E2 + P1F 2 - P1E and F 2 6s6 rounded up to the next integer. 1n1 - 12s21 + 1n2 - 12s22 + g + 1nk - 12s2k where Ri is the sum of the ranks in the ith sample.
B x2a>2 B x21 - a>2 MSE =

n- k

Copyright © 2017 Pearson Education, Inc.

Z05_SULL3539_05_SE_Barrel.indd 1 10/27/15 7:10 PM

 Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­

Chapter 2  Organizing and Summarizing Data • Complement Rule • Factorial Chapter 10  Hypothesis Tests Regarding a Parameter Chapter 14  Inference on the Least-Squares Regression Model and Multiple Regression
frequency • Class midpoint: The sum of consecutive lower class limits
c
P1E 2 = 1 - P1E2 n! = n # 1n - 12 # 1n - 22 # g # 3 # 2 # 1 Test Statistics • Standard Error of the Estimate • Confidence Interval about the Mean Response of y, yn
g 1yi - yn i 2 g residuals
• Relative frequency = divided by 2.
sum of all frequencies • Multiplication Rule for Independent Events pn - p0
2 2
1x* - x2 2
C n g 1xi - x2 2
yn { ta>2 # se
• Permutation of n objects taken r at a time:  x - m0 1
• z0 = • t0 = se = = +
P1E and F 2 = P1E2 # P1F 2 p0 11 - p0 2 s  1n C n- 2 C n- 2
Chapter 3  Numerically Summarizing Data

n!
gxi gxi fi
n Pr = C n • Standard error of b1
• Multiplication Rule for n Independent Events 1n - r2! 1n - 12s2 where x* is the given value of the explanatory variable and

gfi
• x20 =
P1E and F and G g 2 = P1E2 # P1F2 # P1G2 # g
• Population Mean:  m = • Population Mean from Grouped Data:  m = s20 se ta>2 is the critical value with n - 2 degrees of freedom.
N • Combination of n objects taken r at a time: sb1 =

gxi gxi fi
Chapter 11  Inferences on Two Samples 2g 1xi - x2 2 • Prediction Interval about an Individual Response, yn
• Conditional Probability Rule
gfi
n!
• Sample Mean:  x = • Sample Mean from Grouped Data:  x = nC r = • Test Statistic for the Slope of the Least-Squares Regression Line 1x* - x2 2
g 1xi - x2 2
yn { ta>2 # se
n r!1n - r2! • Test Statistic Comparing Two Population Proportions • Test Statistic Comparing Two Means (Independent Sampling) 1
P1E and F 2 N1E and F 2 1+ +
gwi xi
P1F  E2 = (Independent Samples) C

= b1 - b1 b1 - b1 n
• Range = Largest Data Value - Smallest Data Value P1E2 N1E2 1x1 - x2 2 - 1m1 - m2 2 t0 = =
gwi
• Permutations with Repetition: sb1
• Weighted Mean:  xw = pn 1 - pn 2 - 1p1 - p2 2 x1 + x2 t0 =
se n 2g 1xi - x2 2  here x* is the given value of the explanatory variable and
w

• Population Standard Deviation: • General Multiplication Rule n! z0 = where

pn = s21 s22

1 gxi 2 2
n1 + n2 + ta>2 is the critical value with n - 2 degrees of freedom.
n1! # n2! # g # nk!
1 1
P1E and F2 = P1E2 # P1F  E2
gx 2i
• Population Standard Deviation from Grouped Data: 2pn 11 - pn 2 + C n1 n2 • Confidence Interval for the Slope of the Regression Line

g 1xi - m2 2 1 gxi fi 2 2
B n1 n2
gx 2i fi -
- se
gfi
b1 { ta>2 #
g 1xi - m2 fi
N Chapter 6  Discrete Probability Distributions • Confidence Interval for the Difference of Two Means
s = = 2 • Confidence Interval for the Difference of Two Proportions (Independent Samples)
gfi gfi
C N S N 2g 1xi - x2 2
s = = • Mean (Expected Value) of a Discrete Random Variable • Mean and Standard Deviation of a Binomial Random Variable (Independent Samples)

mX = gx # P1x2
B R s21 s22
• Sample Standard Deviation 1x1 - x2 2 { ta>2 where ta>2 is computed with n - 2 degrees of freedom.
1 gxi 2
pn 1 11 - pn 1 2 pn 2 11 - pn 2 2 +
mX = np sX = 2np11 - p2 C n1
gx 2i -
• Sample Standard Deviation from Grouped Data: 1pn 1 - pn 2 2 { za>2 + n2

1 gxi fi 2 2
2

g 1xi - x 2
C n1 n2 Chapter 15  Nonparametric Statistics
gx 2i fi
• Standard Deviation of a Discrete Random Variable • Poisson Probability Distribution Function
gfi
Note: ta>2 is found using the smaller of n1 - 1 or n2 - 1
g 1xi - m2 2fi
2

sX = 3 g 1x - m2 2 # P1x2 = 3 g 3x 2P1x24 - m2X

n
s = = - • Test Statistic for Matched-Pairs Data degrees of freedom. • Test Statistic for a Runs Test for Randomness Large-Sample Case 1 n + 302
B 1 gfi 2 - 1 gfi - 1
C n-1 S n- 1 1lt2 x
s = = P1x2 = e -lt x = 0, 1, 2, c Small-Sample Case  If n1 … 20 and n2 … 20, the test statistic in
R x! d - md • Test Statistic for Comparing Two Population Standard n1n + 12
• Population Variance:  s2 • Binomial Probability Distribution Function t0 = the runs test for randomness is r, the number of runs. T -
sd  1n Deviations 4
x- m • Mean and Standard Deviation of a Poisson Random Variable Large-Sample Case  If n1 7 20 or n2 7 20, the test statistic is z0 =
• Sample Variance:  s2 • Population z-score:  z = P1x2 = nCx px 11 - p2 n - x
s where d is the mean and sd is the standard deviation of the s21 r - mr n1n + 12 12n + 12
• Empirical Rule: If the shape of the distribution is bell- mX = lt sX = 2lt F0 = z0 = where
differenced data. s22 sr C 24
shaped, then x- x Chapter 7  The Normal Distribution
• Sample z-score:  z = where T is the test statistic from the small-sample case.
• Approximately 68% of the data lie within 1 standard s • Confidence Interval for Matched-Pairs Data • Finding a Critical F for the Left Tail 2n1n2 2n1n2 12n1n2 - n2
• Standardizing a Normal Random Variable • Finding the Score:  x = m + zs mr = + 1 and sr =
deviation of the mean sd n B n2 1n - 12 • Test Statistic for the Mann–Whitney Test
d { ta>2 #
• Interquartile Range:  IQR = Q3 - Q1 1
• Approximately 95% of the data lie within 2 standard x- m F1 - a,n1 - 1,n2 - 1 = Small-Sample Case 1 n1 " 20 and n2 " 202
z = 1n Fa,n2 - 1,n1 - 1 • Test Statistic for a One-Sample Sign Test
deviations of the mean Lower fence = Q1 - 1.51IQR2 If S is the sum of the ranks corresponding to the sample from
• Lower and Upper Fences:  s
Small-Sample Case 1n " 252
• Approximately 99.7% of the data lie within 3 standard Upper fence = Q3 + 1.51IQR2 Note: ta>2 is found using n - 1 degrees of freedom. population X, then the test statistic, T, is given by
deviations of the mean Chapter 8  Sampling Distributions Two-Tailed Left-Tailed Right-Tailed
• Five-Number Summary n1 1n1 + 12
Chapter 12  Inference on Categorical Data T = S-
• Mean and Standard Deviation of the Sampling Distribution • Mean and Standard Deviation of the Sampling Distribution H0 : M = M0 H0 : M = M0 H0 : M = M0 2
Minimum, Q1, M, Q3, Maximum
of x of pn • Expected Counts (when testing for goodness of fit) • Chi-Square Test Statistic H1 : M ≠ M0 H1 : M 6 M0 H1 : M 7 M0 Note: The value of S is always obtained by summing the ranks of

x20 = a = a
Chapter 4  Describing the Relation between Two Variables s p11 - p2 Ei = mi = npi for i = 1, 2, c, k 1observed - expected2 2 1Oi - Ei 2 2 The test statistic, k, is the The test statistic, The test statistic, the sample data that correspond to MX in the hypothesis.
mx = m and sx = m pn = p and s pn = smaller of the number of k, is the number k, is the number Large-Sample Case 1 n1 + 202 or 1 n2 + 202
B n
aa s ba s b
xi - x yi - y • Residual = observed y - predicted y = y - yn 2n • Expected Frequencies (when testing for independence or expected Ei
minus signs or plus signs. of plus signs. of minus signs.
homogeneity of proportions) i = 1, 2, c, k n1n2
x y x T -
• Correlation Coefficient:  r = • R2 = r 2 for the least-squares regression model • Sample Proportion:  pn = Large-Sample Case 1n + 252 The test statistic, z0, is 2
n-1 yn = b1x + b0 n 1row total21column total2 All Ei Ú 1 and no more than 20% less than 5. z0 =
Expected frequency = n n1n2 1n1 + n2 + 12
• The equation of the least-squares regression line is 2 Chapter 9  Estimating the Value of a Parameter table total • Test Statistic for Comparing Two Proportions 1k + 0.52 -
• The coefficient of determination, R , measures the 2 B 12
proportion of total variation in the response variable that is (Dependent Samples) z0 =
sy
yn = b1x + b0, where yn is the predicted value, b1 = r # Confidence Intervals 1n • Test Statistic for Spearman’s Rank Correlation Test
explained by the least-squares regression line.
• A 11 - a2 # 100% confidence interval about p is
sx Sample Size 1f12 - f21 2 2 2
is the slope, and b0 = y - b1x is the intercept. • To estimate the population proportion with a margin of x20 = 6gd 2i
pn 11 - pn 2 error E at a 11 - a2 # 100% level of confidence: f12 + f21 where n is the number of minus and plus signs and k is obtained rs = 1 -
pn { za>2 # as described in the small sample case. n1n2 - 12
za>2 2 Chapter 13  Comparing Three or More Means
Chapter 5  Probability B n n = pn 11 - pn 2 a b rounded up to the next integer,
E • Test Statistic for the Wilcoxon Matched-Pairs where di = the difference in the ranks of the two
• Empirical Probability • Addition Rule for Disjoint Events • A 11 - a2 # 100% confidence interval about m is where pn is a prior estimate of the population proportion,
• Test Statistic for One-Way ANOVA • Test Statistic for Tukey’s Test after One-Way ANOVA
Signed-Ranks Test observations in the i th ordered pair.
za>2 2 Mean square due to treatment 1x2 - x1 2 - 1m2 - m1 2 x2 - x1 • Test Statistic for the Kruskal–Wallis Test
x { ta>2 #
s MST Small-Sample Case 1n " 302
frequency of E P1E or F 2 = P1E2 + P1F 2 or n = 0.25 a b rounded up to the next integer when no F = = q = =

a
E

P1E2 ≈ 1n Mean square due to error MSE

s # 12
1 s # 12
1 Two-Tailed Left-Tailed Right-Tailed 12 1 ni 1N + 12 2
number of trials of experiment prior estimate of p is available. a + b a + b H = c Ri - d
• Addition Rule for n Disjoint Events B2 n1 n2 B2 n1 n2 N1N + 12 ni 2
Note: ta>2 is computed using n - 1 degrees of freedom. where H0: MD = 0 H0: MD = 0 H0: MD = 0
• Classical Probability • To estimate the population mean with a margin of error E R21 R22 R2k
P1E or F or G or g 2 = P1E2 + P1F 2 + P1G2 + g
za>2 # s 2
12
• A 11 - a2 # 100% confidence interval about s is
2 2 2

n1 1x1 - x2 + n2 1x2 - x2 + g + nk 1xk - x2 H1: MD ≠ 0 H1: MD 6 0 H0: MD 7 0 = J + + g+ R - 31N + 12

number of ways that E can occur N1E2
• General Addition Rule at a 11 - a2 # 100% level of confidence: n = a b MST =
k- 1 Test Statistic: T is the Test Statistic: Test Statistic:
N1N + 12 n1 n2 nk
P1E2 = = E
number of possible outcomes N1S2 1n - 12s2 1n - 12s2 smaller of T + or T - T = T+ T = T- 
P1E or F 2 = P1E2 + P1F 2 - P1E and F 2 6s6 rounded up to the next integer. 1n1 - 12s21 + 1n2 - 12s22 + g + 1nk - 12s2k where Ri is the sum of the ranks in the ith sample.
B x2a>2 B x21 - a>2 MSE =

n- k

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 Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­

Chapter 2  Organizing and Summarizing Data • Complement Rule • Factorial Chapter 10  Hypothesis Tests Regarding a Parameter Chapter 14  Inference on the Least-Squares Regression Model and Multiple Regression
frequency • Class midpoint: The sum of consecutive lower class limits
c
P1E 2 = 1 - P1E2 n! = n # 1n - 12 # 1n - 22 # g # 3 # 2 # 1 Test Statistics • Standard Error of the Estimate • Confidence Interval about the Mean Response of y, yn
g 1yi - yn i 2 g residuals
• Relative frequency = divided by 2.
sum of all frequencies • Multiplication Rule for Independent Events pn - p0
2 2
1x* - x2 2
C n g 1xi - x2 2
yn { ta>2 # se
• Permutation of n objects taken r at a time:  x - m0 1
• z0 = • t0 = se = = +
P1E and F 2 = P1E2 # P1F 2 p0 11 - p0 2 s  1n C n- 2 C n- 2
Chapter 3  Numerically Summarizing Data

n!
gxi gxi fi
n Pr = C n • Standard error of b1
• Multiplication Rule for n Independent Events 1n - r2! 1n - 12s2 where x* is the given value of the explanatory variable and

gfi
• x20 =
P1E and F and G g 2 = P1E2 # P1F2 # P1G2 # g
• Population Mean:  m = • Population Mean from Grouped Data:  m = s20 se ta>2 is the critical value with n - 2 degrees of freedom.
N • Combination of n objects taken r at a time: sb1 =

gxi gxi fi
Chapter 11  Inferences on Two Samples 2g 1xi - x2 2 • Prediction Interval about an Individual Response, yn
• Conditional Probability Rule
gfi
n!
• Sample Mean:  x = • Sample Mean from Grouped Data:  x = nC r = • Test Statistic for the Slope of the Least-Squares Regression Line 1x* - x2 2
g 1xi - x2 2
yn { ta>2 # se
n r!1n - r2! • Test Statistic Comparing Two Population Proportions • Test Statistic Comparing Two Means (Independent Sampling) 1
P1E and F 2 N1E and F 2 1+ +
gwi xi
P1F  E2 = (Independent Samples) C

= b1 - b1 b1 - b1 n
• Range = Largest Data Value - Smallest Data Value P1E2 N1E2 1x1 - x2 2 - 1m1 - m2 2 t0 = =
gwi
• Permutations with Repetition: sb1
• Weighted Mean:  xw = pn 1 - pn 2 - 1p1 - p2 2 x1 + x2 t0 =
se n 2g 1xi - x2 2  here x* is the given value of the explanatory variable and
w

• Population Standard Deviation: • General Multiplication Rule n! z0 = where

pn = s21 s22

1 gxi 2 2
n1 + n2 + ta>2 is the critical value with n - 2 degrees of freedom.
n1! # n2! # g # nk!
1 1
P1E and F2 = P1E2 # P1F  E2
gx 2i
• Population Standard Deviation from Grouped Data: 2pn 11 - pn 2 + C n1 n2 • Confidence Interval for the Slope of the Regression Line

g 1xi - m2 2 1 gxi fi 2 2
B n1 n2
gx 2i fi -
- se
gfi
b1 { ta>2 #
g 1xi - m2 fi
N Chapter 6  Discrete Probability Distributions • Confidence Interval for the Difference of Two Means
s = = 2 • Confidence Interval for the Difference of Two Proportions (Independent Samples)
gfi gfi
C N S N 2g 1xi - x2 2
s = = • Mean (Expected Value) of a Discrete Random Variable • Mean and Standard Deviation of a Binomial Random Variable (Independent Samples)

mX = gx # P1x2
B R s21 s22
• Sample Standard Deviation 1x1 - x2 2 { ta>2 where ta>2 is computed with n - 2 degrees of freedom.
1 gxi 2
pn 1 11 - pn 1 2 pn 2 11 - pn 2 2 +
mX = np sX = 2np11 - p2 C n1
gx 2i -
• Sample Standard Deviation from Grouped Data: 1pn 1 - pn 2 2 { za>2 + n2

1 gxi fi 2 2
2

g 1xi - x 2
C n1 n2 Chapter 15  Nonparametric Statistics
gx 2i fi
• Standard Deviation of a Discrete Random Variable • Poisson Probability Distribution Function
gfi
Note: ta>2 is found using the smaller of n1 - 1 or n2 - 1
g 1xi - m2 2fi
2

sX = 3 g 1x - m2 2 # P1x2 = 3 g 3x 2P1x24 - m2X

n
s = = - • Test Statistic for Matched-Pairs Data degrees of freedom. • Test Statistic for a Runs Test for Randomness Large-Sample Case 1 n + 302
B 1 gfi 2 - 1 gfi - 1
C n-1 S n- 1 1lt2 x
s = = P1x2 = e -lt x = 0, 1, 2, c Small-Sample Case  If n1 … 20 and n2 … 20, the test statistic in
R x! d - md • Test Statistic for Comparing Two Population Standard n1n + 12
• Population Variance:  s2 • Binomial Probability Distribution Function t0 = the runs test for randomness is r, the number of runs. T -
sd  1n Deviations 4
x- m • Mean and Standard Deviation of a Poisson Random Variable Large-Sample Case  If n1 7 20 or n2 7 20, the test statistic is z0 =
• Sample Variance:  s2 • Population z-score:  z = P1x2 = nCx px 11 - p2 n - x
s where d is the mean and sd is the standard deviation of the s21 r - mr n1n + 12 12n + 12
• Empirical Rule: If the shape of the distribution is bell- mX = lt sX = 2lt F0 = z0 = where
differenced data. s22 sr C 24
shaped, then x- x Chapter 7  The Normal Distribution
• Sample z-score:  z = where T is the test statistic from the small-sample case.
• Approximately 68% of the data lie within 1 standard s • Confidence Interval for Matched-Pairs Data • Finding a Critical F for the Left Tail 2n1n2 2n1n2 12n1n2 - n2
• Standardizing a Normal Random Variable • Finding the Score:  x = m + zs mr = + 1 and sr =
deviation of the mean sd n B n2 1n - 12 • Test Statistic for the Mann–Whitney Test
d { ta>2 #
• Interquartile Range:  IQR = Q3 - Q1 1
• Approximately 95% of the data lie within 2 standard x- m F1 - a,n1 - 1,n2 - 1 = Small-Sample Case 1 n1 " 20 and n2 " 202
z = 1n Fa,n2 - 1,n1 - 1 • Test Statistic for a One-Sample Sign Test
deviations of the mean Lower fence = Q1 - 1.51IQR2 If S is the sum of the ranks corresponding to the sample from
• Lower and Upper Fences:  s
Small-Sample Case 1n " 252
• Approximately 99.7% of the data lie within 3 standard Upper fence = Q3 + 1.51IQR2 Note: ta>2 is found using n - 1 degrees of freedom. population X, then the test statistic, T, is given by
deviations of the mean Chapter 8  Sampling Distributions Two-Tailed Left-Tailed Right-Tailed
• Five-Number Summary n1 1n1 + 12
Chapter 12  Inference on Categorical Data T = S-
• Mean and Standard Deviation of the Sampling Distribution • Mean and Standard Deviation of the Sampling Distribution H0 : M = M0 H0 : M = M0 H0 : M = M0 2
Minimum, Q1, M, Q3, Maximum
of x of pn • Expected Counts (when testing for goodness of fit) • Chi-Square Test Statistic H1 : M ≠ M0 H1 : M 6 M0 H1 : M 7 M0 Note: The value of S is always obtained by summing the ranks of

x20 = a = a
Chapter 4  Describing the Relation between Two Variables s p11 - p2 Ei = mi = npi for i = 1, 2, c, k 1observed - expected2 2 1Oi - Ei 2 2 The test statistic, k, is the The test statistic, The test statistic, the sample data that correspond to MX in the hypothesis.
mx = m and sx = m pn = p and s pn = smaller of the number of k, is the number k, is the number Large-Sample Case 1 n1 + 202 or 1 n2 + 202
B n
aa s ba s b
xi - x yi - y • Residual = observed y - predicted y = y - yn 2n • Expected Frequencies (when testing for independence or expected Ei
minus signs or plus signs. of plus signs. of minus signs.
homogeneity of proportions) i = 1, 2, c, k n1n2
x y x T -
• Correlation Coefficient:  r = • R2 = r 2 for the least-squares regression model • Sample Proportion:  pn = Large-Sample Case 1n + 252 The test statistic, z0, is 2
n-1 yn = b1x + b0 n 1row total21column total2 All Ei Ú 1 and no more than 20% less than 5. z0 =
Expected frequency = n n1n2 1n1 + n2 + 12
• The equation of the least-squares regression line is 2 Chapter 9  Estimating the Value of a Parameter table total • Test Statistic for Comparing Two Proportions 1k + 0.52 -
• The coefficient of determination, R , measures the 2 B 12
proportion of total variation in the response variable that is (Dependent Samples) z0 =
sy
yn = b1x + b0, where yn is the predicted value, b1 = r # Confidence Intervals 1n • Test Statistic for Spearman’s Rank Correlation Test
explained by the least-squares regression line.
• A 11 - a2 # 100% confidence interval about p is
sx Sample Size 1f12 - f21 2 2 2
is the slope, and b0 = y - b1x is the intercept. • To estimate the population proportion with a margin of x20 = 6gd 2i
pn 11 - pn 2 error E at a 11 - a2 # 100% level of confidence: f12 + f21 where n is the number of minus and plus signs and k is obtained rs = 1 -
pn { za>2 # as described in the small sample case. n1n2 - 12
za>2 2 Chapter 13  Comparing Three or More Means
Chapter 5  Probability B n n = pn 11 - pn 2 a b rounded up to the next integer,
E • Test Statistic for the Wilcoxon Matched-Pairs where di = the difference in the ranks of the two
• Empirical Probability • Addition Rule for Disjoint Events • A 11 - a2 # 100% confidence interval about m is where pn is a prior estimate of the population proportion,
• Test Statistic for One-Way ANOVA • Test Statistic for Tukey’s Test after One-Way ANOVA
Signed-Ranks Test observations in the i th ordered pair.
za>2 2 Mean square due to treatment 1x2 - x1 2 - 1m2 - m1 2 x2 - x1 • Test Statistic for the Kruskal–Wallis Test
x { ta>2 #
s MST Small-Sample Case 1n " 302
frequency of E P1E or F 2 = P1E2 + P1F 2 or n = 0.25 a b rounded up to the next integer when no F = = q = =

a
E

P1E2 ≈ 1n Mean square due to error MSE

s # 12
1 s # 12
1 Two-Tailed Left-Tailed Right-Tailed 12 1 ni 1N + 12 2
number of trials of experiment prior estimate of p is available. a + b a + b H = c Ri - d
• Addition Rule for n Disjoint Events B2 n1 n2 B2 n1 n2 N1N + 12 ni 2
Note: ta>2 is computed using n - 1 degrees of freedom. where H0: MD = 0 H0: MD = 0 H0: MD = 0
• Classical Probability • To estimate the population mean with a margin of error E R21 R22 R2k
P1E or F or G or g 2 = P1E2 + P1F 2 + P1G2 + g
za>2 # s 2
12
• A 11 - a2 # 100% confidence interval about s is
2 2 2

n1 1x1 - x2 + n2 1x2 - x2 + g + nk 1xk - x2 H1: MD ≠ 0 H1: MD 6 0 H0: MD 7 0 = J + + g+ R - 31N + 12

number of ways that E can occur N1E2
• General Addition Rule at a 11 - a2 # 100% level of confidence: n = a b MST =
k- 1 Test Statistic: T is the Test Statistic: Test Statistic:
N1N + 12 n1 n2 nk
P1E2 = = E
number of possible outcomes N1S2 1n - 12s2 1n - 12s2 smaller of T + or T - T = T+ T = T- 
P1E or F 2 = P1E2 + P1F 2 - P1E and F 2 6s6 rounded up to the next integer. 1n1 - 12s21 + 1n2 - 12s22 + g + 1nk - 12s2k where Ri is the sum of the ranks in the ith sample.
B x2a>2 B x21 - a>2 MSE =

n- k

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 Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­

Table I Table V Table VIII

Standard Normal Distribution
Random Numbers z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 Chi-Square (X2) Distribution
Column Number Area
Area in Area to the Right of Critical Value
Row 23.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002
right tail
Degrees of
23.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003
Number 01–05 06–10 11–15 16–20 21–25 26–30 31–35 36–40 41–45 46–50 z 23.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005 Freedom 0.995 0.99 0.975 0.95 0.90 0.10 0.05 0.025 0.01 0.005
23.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007
01 89392 23212 74483 36590 25956 36544 68518 40805 09980 00467 23.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010
t 1 — — 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7.879
02 61458 17639 96252 95649 73727 33912 72896 66218 52341 97141 22.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014 2 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.597
03 11452 74197 81962 48443 90360 26480 73231 37740 26628 44690 22.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019 3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838
04 27575 04429 31308 02241 01698 19191 18948 78871 36030 23980 22.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026 Table VII 4 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860
22.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
05 36829 59109 88976 46845 28329 47460 88944 08264 00843 84592 22.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048 t-Distribution 5 0.412 0.554 0.831 1.145 1.610 9.236 11.070 12.833 15.086 16.750
06 81902 93458 42161 26099 09419 89073 82849 09160 61845 40906 22.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064
Area in Right Tail 6 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548
07 59761 55212 33360 68751 86737 79743 85262 31887 37879 17525 22.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
22.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
7 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278
08 46827 25906 64708 20307 78423 15910 86548 08763 47050 18513 22.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143 df 0.25 0.20 0.15 0.10 0.05 0.025 0.02 0.01 0.005 0.0025 0.001 0.0005 8 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955
09 24040 66449 32353 83668 13874 86741 81312 54185 78824 00718 22.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183 9 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589
10 98144 96372 50277 15571 82261 66628 31457 00377 63423 55141 21.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233 1 1.000 1.376 1.963 3.078 6.314 12.706 15.894 31.821 63.657 127.321 318.309 636.619 10 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188
21.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294 2 0.816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 9.925 14.089 22.327 31.599
11 14228 17930 30118 00438 49666 65189 62869 31304 17117 71489 21.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367 11 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.757
12 55366 51057 90065 14791 62426 02957 85518 28822 30588 32798 21.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 3 0.765 0.978 1.250 1.638 2.353 3.182 3.482 4.541 5.841 7.453 10.215 12.924 12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.300
13 96101 30646 35526 90389 73634 79304 96635 06626 94683 16696 21.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559 4 0.741 0.941 1.190 1.533 2.132 2.776 2.999 3.747 4.604 5.598 7.173 8.610 13 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819
14 38152 55474 30153 26525 83647 31988 82182 98377 33802 80471 21.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681 5 0.727 0.920 1.156 1.476 2.015 2.571 2.757 3.365 4.032 4.773 5.893 6.869 14 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319
15 85007 18416 24661 95581 45868 15662 28906 36392 07617 50248 21.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 6 0.718 0.906 1.134 1.440 1.943 2.447 2.612 3.143 3.707 4.317 5.208 5.959
21.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
15 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801
16 85544 15890 80011 18160 33468 84106 40603 01315 74664 20553 21.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170 7 0.711 0.896 1.119 1.415 1.895 2.365 2.517 2.998 3.499 4.029 4.785 5.408 16 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267
17 10446 20699 98370 17684 16932 80449 92654 02084 19985 59321 21.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379 8 0.706 0.889 1.108 1.397 1.860 2.306 2.449 2.896 3.355 3.833 4.501 5.041 17 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718
18 67237 45509 17638 65115 29757 80705 82686 48565 72612 61760 20.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611 9 0.703 0.883 1.100 1.383 1.833 2.262 2.398 2.821 3.250 3.690 4.297 4.781 18 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156
19 23026 89817 05403 82209 30573 47501 00135 33955 50250 72592
20.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867 10 0.700 0.879 1.093 1.372 1.812 2.228 2.359 2.764 3.169 3.581 4.144 4.587 19 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582
20.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148
20 67411 58542 18678 46491 13219 84084 27783 34508 55158 78742 20.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451 11 0.697 0.876 1.088 1.363 1.796 2.201 2.328 2.718 3.106 3.497 4.025 4.437 20 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997
20.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776 12 0.695 0.873 1.083 1.356 1.782 2.179 2.303 2.681 3.055 3.428 3.930 4.318
21 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401
20.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121 13 0.694 0.870 1.079 1.350 1.771 2.160 2.282 2.650 3.012 3.372 3.852 4.221
Table II 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483 22 8.643 9.542 10.982 12.338 14.041 30.813 33.924 36.781 40.289 42.796
20.3 14 0.692 0.868 1.076 1.345 1.761 2.145 2.264 2.624 2.977 3.326 3.787 4.140
20.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859 23 9.260 10.196 11.689 13.091 14.848 32.007 35.172 38.076 41.638 44.181
Critical Values (CV) for Correlation Coefficient 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247 15 0.691 0.866 1.074 1.341 1.753 2.131 2.249 2.602 2.947 3.286 3.733 4.073
20.1 24 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.559
20.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641 16 0.690 0.865 1.071 1.337 1.746 2.120 2.235 2.583 2.921 3.252 3.686 4.015 25 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928
n CV n CV n CV n CV 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
17 0.689 0.863 1.069 1.333 1.740 2.110 2.224 2.567 2.898 3.222 3.646 3.965
3 0.997 10 0.632 17 0.482 24 0.404 18 0.688 0.862 1.067 1.330 1.734 2.101 2.214 2.552 2.878 3.197 3.610 3.922 26 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 19 0.688 0.861 1.066 1.328 1.729 2.093 2.205 2.539 2.861 3.174 3.579 3.883 27 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.195 46.963 49.645
4 0.950 11 0.602 18 0.468 25 0.396 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 28 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993
20 0.687 0.860 1.064 1.325 1.725 2.086 2.197 2.528 2.845 3.153 3.552 3.850
5 0.878 12 0.576 19 0.456 26 0.388 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 29 13.121 14.256 16.047 17.708 19.768 39.087 42.557 45.722 49.588 52.336
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 21 0.686 0.859 1.063 1.323 1.721 2.080 2.189 2.518 2.831 3.135 3.527 3.819 30 13.787 14.953 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672
6 0.811 13 0.553 20 0.444 27 0.381 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 22 0.686 0.858 1.061 1.321 1.717 2.074 2.183 2.508 2.819 3.119 3.505 3.792
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 23 0.685 0.858 1.060 1.319 1.714 2.069 2.177 2.500 2.807 3.104 3.485 3.768 40 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66.766
7 0.754 14 0.532 21 0.433 28 0.374 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
24 0.685 0.857 1.059 1.318 1.711 2.064 2.172 2.492 2.797 3.091 3.467 3.745 50 27.991 29.707 32.357 34.764 37.689 63.167 67.505 71.420 76.154 79.490
8 0.707 15 0.514 22 0.423 29 0.367 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 25 0.684 0.856 1.058 1.316 1.708 2.060 2.167 2.485 2.787 3.078 3.450 3.725 60 35.534 37.485 40.482 43.188 46.459 74.397 79.082 83.298 88.379 91.952
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 70 43.275 45.442 48.758 51.739 55.329 85.527 90.531 95.023 100.425 104.215
9 0.666 16 0.497 23 0.413 30 0.361 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 26 0.684 0.856 1.058 1.315 1.706 2.056 2.162 2.479 2.779 3.067 3.435 3.707
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 80 51.172 53.540 57.153 60.391 64.278 96.578 101.879 106.629 112.329 116.321
27 0.684 0.855 1.057 1.314 1.703 2.052 2.158 2.473 2.771 3.057 3.421 3.690
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 90 59.196 61.754 65.647 69.126 73.291 107.565 113.145 118.136 124.116 128.299
28 0.683 0.855 1.056 1.313 1.701 2.048 2.154 2.467 2.763 3.047 3.408 3.674
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 100 67.328 70.065 74.222 77.929 82.358 118.498 124.342 129.561 135.807 140.169
Table VI  1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
29 0.683 0.854 1.055 1.311 1.699 2.045 2.150 2.462 2.756 3.038 3.396 3.659
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 30 0.683 0.854 1.055 1.310 1.697 2.042 2.147 2.457 2.750 3.030 3.385 3.646
Critical Values for Normal Probability Plots 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
31 0.682 0.853 1.054 1.309 1.696 2.040 2.144 2.453 2.744 3.022 3.375 3.633
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
Sample Size, n Critical Value Sample Size, n Critical Value Sample Size, n Critical Value 32 0.682 0.853 1.054 1.309 1.694 2.037 2.141 2.449 2.738 3.015 3.365 3.622 Right tail Left tail Two tails
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 33 0.682 0.853 1.053 1.308 1.692 2.035 2.138 2.445 2.733 3.008 3.356 3.611 Area 5 12a
 5 0.880 13 0.932 21 0.952
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 34 0.682 0.852 1.052 1.307 1.691 2.032 2.136 2.441 2.728 3.002 3.348 3.601
 6 0.888 14 0.935 22 0.954 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 35 0.682 0.852 1.052 1.306 1.690 2.030 2.133 2.438 2.724 2.996 3.340 3.591
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
 7 0.898 15 0.939 23 0.956 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
36 0.681 0.852 1.052 1.306 1.688 2.028 2.131 2.434 2.719 2.990 3.333 3.582
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 37 0.681 0.851 1.051 1.305 1.687 2.026 2.129 2.431 2.715 2.985 3.326 3.574 a
 8 0.906 16 0.941 24 0.957 a a
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 38 0.681 0.851 1.051 1.304 1.686 2.024 2.127 2.429 2.712 2.980 3.319 3.566 a
 9 0.912 17 0.944 25 0.959 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2 2
39 0.681 0.851 1.050 1.304 1.685 2.023 2.125 2.426 2.708 2.976 3.313 3.558
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
10 0.918 18 0.946 30 0.960 40 0.681 0.851 1.050 1.303 1.684 2.021 2.123 2.423 2.704 2.971 3.307 3.551 Xa2 The area to the 2
2
X12a Xa2 The area to the right
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 X12a The area to the
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 50 0.679 0.849 1.047 1.299 1.676 2.009 2.109 2.403 2.678 2.937 3.261 3.496 right of this 2 2 of this value is a .
11 0.923 19 0.949 value is a. right of this 2
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 60 0.679 0.848 1.045 1.296 1.671 2.000 2.099 2.390 2.660 2.915 3.232 3.460 value is 12a. The area to the right
12 0.928 20 0.951 3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 70 0.678 0.847 1.044 1.294 1.667 1.994 2.093 2.381 2.648 2.899 3.211 3.435 a.
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998 of this value is 12
80 0.678 0.846 1.043 1.292 1.664 1.990 2.088 2.374 2.639 2.887 3.195 3.416 2
Source: S. W. Looney and T. R. Gulledge, Jr. “Use of the ­Correlation Coefficient with Normal Probability Plots,” ­American ­Statistician 39(Feb. 1985): 75–79.
90 0.677 0.846 1.042 1.291 1.662 1.987 2.084 2.368 2.632 2.878 3.183 3.402
Table  VII   Confidence Interval Critical Values, zA/2 Hypothesis Testing Critical Values
100 0.677 0.845 1.042 1.290 1.660 1.984 2.081 2.364 2.626 2.871 3.174 3.390
Level of Confidence Critical Value, zA/2 Level of Significance, Left-Tailed Right- Two- 1000 0.675 0.842 1.037 1.282 1.646 1.962 2.056 2.330 2.581 2.813 3.098 3.300
A Tailed Tailed z 0.674 0.842 1.036 1.282 1.645 1.960 2.054 2.326 2.576 2.807 3.090 3.291
0.90 or 90% 1.645
0.95 or 95% 1.96 0.10 - 1.28 1.28 { 1.645

0.98 or 98% 2.33 0.05 - 1.645 1.645 { 1.96

0.99 or 99% 2.575 0.01 - 2.33 2.33 { 2.575

Copyright © 2017 Pearson Education, Inc.

Z05_SULL3539_05_SE_Barrel.indd 2 10/27/15 7:10 PM

 Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­

Table I Table V Table VIII

Standard Normal Distribution
Random Numbers z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 Chi-Square (X2) Distribution
Column Number Area
Area in Area to the Right of Critical Value
Row 23.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002
right tail
Degrees of
23.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003
Number 01–05 06–10 11–15 16–20 21–25 26–30 31–35 36–40 41–45 46–50 z 23.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005 Freedom 0.995 0.99 0.975 0.95 0.90 0.10 0.05 0.025 0.01 0.005
23.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007
01 89392 23212 74483 36590 25956 36544 68518 40805 09980 00467 23.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010
t 1 — — 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7.879
02 61458 17639 96252 95649 73727 33912 72896 66218 52341 97141 22.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014 2 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.597
03 11452 74197 81962 48443 90360 26480 73231 37740 26628 44690 22.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019 3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838
04 27575 04429 31308 02241 01698 19191 18948 78871 36030 23980 22.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026 Table VII 4 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860
22.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
05 36829 59109 88976 46845 28329 47460 88944 08264 00843 84592 22.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048 t-Distribution 5 0.412 0.554 0.831 1.145 1.610 9.236 11.070 12.833 15.086 16.750
06 81902 93458 42161 26099 09419 89073 82849 09160 61845 40906 22.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064
Area in Right Tail 6 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548
07 59761 55212 33360 68751 86737 79743 85262 31887 37879 17525 22.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
22.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
7 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278
08 46827 25906 64708 20307 78423 15910 86548 08763 47050 18513 22.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143 df 0.25 0.20 0.15 0.10 0.05 0.025 0.02 0.01 0.005 0.0025 0.001 0.0005 8 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955
09 24040 66449 32353 83668 13874 86741 81312 54185 78824 00718 22.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183 9 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589
10 98144 96372 50277 15571 82261 66628 31457 00377 63423 55141 21.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233 1 1.000 1.376 1.963 3.078 6.314 12.706 15.894 31.821 63.657 127.321 318.309 636.619 10 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188
21.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294 2 0.816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 9.925 14.089 22.327 31.599
11 14228 17930 30118 00438 49666 65189 62869 31304 17117 71489 21.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367 11 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.757
12 55366 51057 90065 14791 62426 02957 85518 28822 30588 32798 21.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 3 0.765 0.978 1.250 1.638 2.353 3.182 3.482 4.541 5.841 7.453 10.215 12.924 12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.300
13 96101 30646 35526 90389 73634 79304 96635 06626 94683 16696 21.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559 4 0.741 0.941 1.190 1.533 2.132 2.776 2.999 3.747 4.604 5.598 7.173 8.610 13 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819
14 38152 55474 30153 26525 83647 31988 82182 98377 33802 80471 21.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681 5 0.727 0.920 1.156 1.476 2.015 2.571 2.757 3.365 4.032 4.773 5.893 6.869 14 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319
15 85007 18416 24661 95581 45868 15662 28906 36392 07617 50248 21.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 6 0.718 0.906 1.134 1.440 1.943 2.447 2.612 3.143 3.707 4.317 5.208 5.959
21.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
15 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801
16 85544 15890 80011 18160 33468 84106 40603 01315 74664 20553 21.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170 7 0.711 0.896 1.119 1.415 1.895 2.365 2.517 2.998 3.499 4.029 4.785 5.408 16 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267
17 10446 20699 98370 17684 16932 80449 92654 02084 19985 59321 21.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379 8 0.706 0.889 1.108 1.397 1.860 2.306 2.449 2.896 3.355 3.833 4.501 5.041 17 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718
18 67237 45509 17638 65115 29757 80705 82686 48565 72612 61760 20.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611 9 0.703 0.883 1.100 1.383 1.833 2.262 2.398 2.821 3.250 3.690 4.297 4.781 18 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156
19 23026 89817 05403 82209 30573 47501 00135 33955 50250 72592
20.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867 10 0.700 0.879 1.093 1.372 1.812 2.228 2.359 2.764 3.169 3.581 4.144 4.587 19 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582
20.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148
20 67411 58542 18678 46491 13219 84084 27783 34508 55158 78742 20.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451 11 0.697 0.876 1.088 1.363 1.796 2.201 2.328 2.718 3.106 3.497 4.025 4.437 20 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997
20.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776 12 0.695 0.873 1.083 1.356 1.782 2.179 2.303 2.681 3.055 3.428 3.930 4.318
21 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401
20.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121 13 0.694 0.870 1.079 1.350 1.771 2.160 2.282 2.650 3.012 3.372 3.852 4.221
Table II 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483 22 8.643 9.542 10.982 12.338 14.041 30.813 33.924 36.781 40.289 42.796
20.3 14 0.692 0.868 1.076 1.345 1.761 2.145 2.264 2.624 2.977 3.326 3.787 4.140
20.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859 23 9.260 10.196 11.689 13.091 14.848 32.007 35.172 38.076 41.638 44.181
Critical Values (CV) for Correlation Coefficient 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247 15 0.691 0.866 1.074 1.341 1.753 2.131 2.249 2.602 2.947 3.286 3.733 4.073
20.1 24 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.559
20.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641 16 0.690 0.865 1.071 1.337 1.746 2.120 2.235 2.583 2.921 3.252 3.686 4.015 25 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928
n CV n CV n CV n CV 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
17 0.689 0.863 1.069 1.333 1.740 2.110 2.224 2.567 2.898 3.222 3.646 3.965
3 0.997 10 0.632 17 0.482 24 0.404 18 0.688 0.862 1.067 1.330 1.734 2.101 2.214 2.552 2.878 3.197 3.610 3.922 26 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 19 0.688 0.861 1.066 1.328 1.729 2.093 2.205 2.539 2.861 3.174 3.579 3.883 27 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.195 46.963 49.645
4 0.950 11 0.602 18 0.468 25 0.396 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 28 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993
20 0.687 0.860 1.064 1.325 1.725 2.086 2.197 2.528 2.845 3.153 3.552 3.850
5 0.878 12 0.576 19 0.456 26 0.388 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 29 13.121 14.256 16.047 17.708 19.768 39.087 42.557 45.722 49.588 52.336
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 21 0.686 0.859 1.063 1.323 1.721 2.080 2.189 2.518 2.831 3.135 3.527 3.819 30 13.787 14.953 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672
6 0.811 13 0.553 20 0.444 27 0.381 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 22 0.686 0.858 1.061 1.321 1.717 2.074 2.183 2.508 2.819 3.119 3.505 3.792
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 23 0.685 0.858 1.060 1.319 1.714 2.069 2.177 2.500 2.807 3.104 3.485 3.768 40 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66.766
7 0.754 14 0.532 21 0.433 28 0.374 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
24 0.685 0.857 1.059 1.318 1.711 2.064 2.172 2.492 2.797 3.091 3.467 3.745 50 27.991 29.707 32.357 34.764 37.689 63.167 67.505 71.420 76.154 79.490
8 0.707 15 0.514 22 0.423 29 0.367 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 25 0.684 0.856 1.058 1.316 1.708 2.060 2.167 2.485 2.787 3.078 3.450 3.725 60 35.534 37.485 40.482 43.188 46.459 74.397 79.082 83.298 88.379 91.952
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 70 43.275 45.442 48.758 51.739 55.329 85.527 90.531 95.023 100.425 104.215
9 0.666 16 0.497 23 0.413 30 0.361 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 26 0.684 0.856 1.058 1.315 1.706 2.056 2.162 2.479 2.779 3.067 3.435 3.707
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 80 51.172 53.540 57.153 60.391 64.278 96.578 101.879 106.629 112.329 116.321
27 0.684 0.855 1.057 1.314 1.703 2.052 2.158 2.473 2.771 3.057 3.421 3.690
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 90 59.196 61.754 65.647 69.126 73.291 107.565 113.145 118.136 124.116 128.299
28 0.683 0.855 1.056 1.313 1.701 2.048 2.154 2.467 2.763 3.047 3.408 3.674
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 100 67.328 70.065 74.222 77.929 82.358 118.498 124.342 129.561 135.807 140.169
Table VI  1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
29 0.683 0.854 1.055 1.311 1.699 2.045 2.150 2.462 2.756 3.038 3.396 3.659
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 30 0.683 0.854 1.055 1.310 1.697 2.042 2.147 2.457 2.750 3.030 3.385 3.646
Critical Values for Normal Probability Plots 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
31 0.682 0.853 1.054 1.309 1.696 2.040 2.144 2.453 2.744 3.022 3.375 3.633
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
Sample Size, n Critical Value Sample Size, n Critical Value Sample Size, n Critical Value 32 0.682 0.853 1.054 1.309 1.694 2.037 2.141 2.449 2.738 3.015 3.365 3.622 Right tail Left tail Two tails
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 33 0.682 0.853 1.053 1.308 1.692 2.035 2.138 2.445 2.733 3.008 3.356 3.611 Area 5 12a
 5 0.880 13 0.932 21 0.952
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 34 0.682 0.852 1.052 1.307 1.691 2.032 2.136 2.441 2.728 3.002 3.348 3.601
 6 0.888 14 0.935 22 0.954 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 35 0.682 0.852 1.052 1.306 1.690 2.030 2.133 2.438 2.724 2.996 3.340 3.591
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
 7 0.898 15 0.939 23 0.956 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
36 0.681 0.852 1.052 1.306 1.688 2.028 2.131 2.434 2.719 2.990 3.333 3.582
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 37 0.681 0.851 1.051 1.305 1.687 2.026 2.129 2.431 2.715 2.985 3.326 3.574 a
 8 0.906 16 0.941 24 0.957 a a
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 38 0.681 0.851 1.051 1.304 1.686 2.024 2.127 2.429 2.712 2.980 3.319 3.566 a
 9 0.912 17 0.944 25 0.959 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2 2
39 0.681 0.851 1.050 1.304 1.685 2.023 2.125 2.426 2.708 2.976 3.313 3.558
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
10 0.918 18 0.946 30 0.960 40 0.681 0.851 1.050 1.303 1.684 2.021 2.123 2.423 2.704 2.971 3.307 3.551 Xa2 The area to the 2
2
X12a Xa2 The area to the right
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 X12a The area to the
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 50 0.679 0.849 1.047 1.299 1.676 2.009 2.109 2.403 2.678 2.937 3.261 3.496 right of this 2 2 of this value is a .
11 0.923 19 0.949 value is a. right of this 2
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 60 0.679 0.848 1.045 1.296 1.671 2.000 2.099 2.390 2.660 2.915 3.232 3.460 value is 12a. The area to the right
12 0.928 20 0.951 3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 70 0.678 0.847 1.044 1.294 1.667 1.994 2.093 2.381 2.648 2.899 3.211 3.435 a.
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998 of this value is 12
80 0.678 0.846 1.043 1.292 1.664 1.990 2.088 2.374 2.639 2.887 3.195 3.416 2
Source: S. W. Looney and T. R. Gulledge, Jr. “Use of the ­Correlation Coefficient with Normal Probability Plots,” ­American ­Statistician 39(Feb. 1985): 75–79.
90 0.677 0.846 1.042 1.291 1.662 1.987 2.084 2.368 2.632 2.878 3.183 3.402
Table  VII   Confidence Interval Critical Values, zA/2 Hypothesis Testing Critical Values
100 0.677 0.845 1.042 1.290 1.660 1.984 2.081 2.364 2.626 2.871 3.174 3.390
Level of Confidence Critical Value, zA/2 Level of Significance, Left-Tailed Right- Two- 1000 0.675 0.842 1.037 1.282 1.646 1.962 2.056 2.330 2.581 2.813 3.098 3.300
A Tailed Tailed z 0.674 0.842 1.036 1.282 1.645 1.960 2.054 2.326 2.576 2.807 3.090 3.291
0.90 or 90% 1.645
0.95 or 95% 1.96 0.10 - 1.28 1.28 { 1.645

0.98 or 98% 2.33 0.05 - 1.645 1.645 { 1.96

0.99 or 99% 2.575 0.01 - 2.33 2.33 { 2.575

Copyright © 2017 Pearson Education, Inc.

Z05_SULL3539_05_SE_Barrel.indd 2 10/27/15 7:10 PM

 Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­

Table I Table V Table VIII

Standard Normal Distribution
Random Numbers z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 Chi-Square (X2) Distribution
Column Number Area
Area in Area to the Right of Critical Value
Row 23.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002
right tail
Degrees of
23.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003
Number 01–05 06–10 11–15 16–20 21–25 26–30 31–35 36–40 41–45 46–50 z 23.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005 Freedom 0.995 0.99 0.975 0.95 0.90 0.10 0.05 0.025 0.01 0.005
23.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007
01 89392 23212 74483 36590 25956 36544 68518 40805 09980 00467 23.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010
t 1 — — 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7.879
02 61458 17639 96252 95649 73727 33912 72896 66218 52341 97141 22.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014 2 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.597
03 11452 74197 81962 48443 90360 26480 73231 37740 26628 44690 22.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019 3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838
04 27575 04429 31308 02241 01698 19191 18948 78871 36030 23980 22.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026 Table VII 4 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860
22.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
05 36829 59109 88976 46845 28329 47460 88944 08264 00843 84592 22.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048 t-Distribution 5 0.412 0.554 0.831 1.145 1.610 9.236 11.070 12.833 15.086 16.750
06 81902 93458 42161 26099 09419 89073 82849 09160 61845 40906 22.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064
Area in Right Tail 6 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548
07 59761 55212 33360 68751 86737 79743 85262 31887 37879 17525 22.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
22.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
7 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278
08 46827 25906 64708 20307 78423 15910 86548 08763 47050 18513 22.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143 df 0.25 0.20 0.15 0.10 0.05 0.025 0.02 0.01 0.005 0.0025 0.001 0.0005 8 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955
09 24040 66449 32353 83668 13874 86741 81312 54185 78824 00718 22.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183 9 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589
10 98144 96372 50277 15571 82261 66628 31457 00377 63423 55141 21.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233 1 1.000 1.376 1.963 3.078 6.314 12.706 15.894 31.821 63.657 127.321 318.309 636.619 10 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188
21.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294 2 0.816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 9.925 14.089 22.327 31.599
11 14228 17930 30118 00438 49666 65189 62869 31304 17117 71489 21.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367 11 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.757
12 55366 51057 90065 14791 62426 02957 85518 28822 30588 32798 21.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 3 0.765 0.978 1.250 1.638 2.353 3.182 3.482 4.541 5.841 7.453 10.215 12.924 12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.300
13 96101 30646 35526 90389 73634 79304 96635 06626 94683 16696 21.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559 4 0.741 0.941 1.190 1.533 2.132 2.776 2.999 3.747 4.604 5.598 7.173 8.610 13 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819
14 38152 55474 30153 26525 83647 31988 82182 98377 33802 80471 21.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681 5 0.727 0.920 1.156 1.476 2.015 2.571 2.757 3.365 4.032 4.773 5.893 6.869 14 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319
15 85007 18416 24661 95581 45868 15662 28906 36392 07617 50248 21.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 6 0.718 0.906 1.134 1.440 1.943 2.447 2.612 3.143 3.707 4.317 5.208 5.959
21.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
15 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801
16 85544 15890 80011 18160 33468 84106 40603 01315 74664 20553 21.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170 7 0.711 0.896 1.119 1.415 1.895 2.365 2.517 2.998 3.499 4.029 4.785 5.408 16 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267
17 10446 20699 98370 17684 16932 80449 92654 02084 19985 59321 21.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379 8 0.706 0.889 1.108 1.397 1.860 2.306 2.449 2.896 3.355 3.833 4.501 5.041 17 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718
18 67237 45509 17638 65115 29757 80705 82686 48565 72612 61760 20.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611 9 0.703 0.883 1.100 1.383 1.833 2.262 2.398 2.821 3.250 3.690 4.297 4.781 18 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156
19 23026 89817 05403 82209 30573 47501 00135 33955 50250 72592
20.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867 10 0.700 0.879 1.093 1.372 1.812 2.228 2.359 2.764 3.169 3.581 4.144 4.587 19 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582
20.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148
20 67411 58542 18678 46491 13219 84084 27783 34508 55158 78742 20.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451 11 0.697 0.876 1.088 1.363 1.796 2.201 2.328 2.718 3.106 3.497 4.025 4.437 20 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997
20.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776 12 0.695 0.873 1.083 1.356 1.782 2.179 2.303 2.681 3.055 3.428 3.930 4.318
21 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401
20.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121 13 0.694 0.870 1.079 1.350 1.771 2.160 2.282 2.650 3.012 3.372 3.852 4.221
Table II 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483 22 8.643 9.542 10.982 12.338 14.041 30.813 33.924 36.781 40.289 42.796
20.3 14 0.692 0.868 1.076 1.345 1.761 2.145 2.264 2.624 2.977 3.326 3.787 4.140
20.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859 23 9.260 10.196 11.689 13.091 14.848 32.007 35.172 38.076 41.638 44.181
Critical Values (CV) for Correlation Coefficient 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247 15 0.691 0.866 1.074 1.341 1.753 2.131 2.249 2.602 2.947 3.286 3.733 4.073
20.1 24 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.559
20.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641 16 0.690 0.865 1.071 1.337 1.746 2.120 2.235 2.583 2.921 3.252 3.686 4.015 25 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928
n CV n CV n CV n CV 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
17 0.689 0.863 1.069 1.333 1.740 2.110 2.224 2.567 2.898 3.222 3.646 3.965
3 0.997 10 0.632 17 0.482 24 0.404 18 0.688 0.862 1.067 1.330 1.734 2.101 2.214 2.552 2.878 3.197 3.610 3.922 26 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 19 0.688 0.861 1.066 1.328 1.729 2.093 2.205 2.539 2.861 3.174 3.579 3.883 27 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.195 46.963 49.645
4 0.950 11 0.602 18 0.468 25 0.396 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 28 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993
20 0.687 0.860 1.064 1.325 1.725 2.086 2.197 2.528 2.845 3.153 3.552 3.850
5 0.878 12 0.576 19 0.456 26 0.388 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 29 13.121 14.256 16.047 17.708 19.768 39.087 42.557 45.722 49.588 52.336
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 21 0.686 0.859 1.063 1.323 1.721 2.080 2.189 2.518 2.831 3.135 3.527 3.819 30 13.787 14.953 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672
6 0.811 13 0.553 20 0.444 27 0.381 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 22 0.686 0.858 1.061 1.321 1.717 2.074 2.183 2.508 2.819 3.119 3.505 3.792
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 23 0.685 0.858 1.060 1.319 1.714 2.069 2.177 2.500 2.807 3.104 3.485 3.768 40 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66.766
7 0.754 14 0.532 21 0.433 28 0.374 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
24 0.685 0.857 1.059 1.318 1.711 2.064 2.172 2.492 2.797 3.091 3.467 3.745 50 27.991 29.707 32.357 34.764 37.689 63.167 67.505 71.420 76.154 79.490
8 0.707 15 0.514 22 0.423 29 0.367 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 25 0.684 0.856 1.058 1.316 1.708 2.060 2.167 2.485 2.787 3.078 3.450 3.725 60 35.534 37.485 40.482 43.188 46.459 74.397 79.082 83.298 88.379 91.952
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 70 43.275 45.442 48.758 51.739 55.329 85.527 90.531 95.023 100.425 104.215
9 0.666 16 0.497 23 0.413 30 0.361 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 26 0.684 0.856 1.058 1.315 1.706 2.056 2.162 2.479 2.779 3.067 3.435 3.707
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 80 51.172 53.540 57.153 60.391 64.278 96.578 101.879 106.629 112.329 116.321
27 0.684 0.855 1.057 1.314 1.703 2.052 2.158 2.473 2.771 3.057 3.421 3.690
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 90 59.196 61.754 65.647 69.126 73.291 107.565 113.145 118.136 124.116 128.299
28 0.683 0.855 1.056 1.313 1.701 2.048 2.154 2.467 2.763 3.047 3.408 3.674
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 100 67.328 70.065 74.222 77.929 82.358 118.498 124.342 129.561 135.807 140.169
Table VI  1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
29 0.683 0.854 1.055 1.311 1.699 2.045 2.150 2.462 2.756 3.038 3.396 3.659
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 30 0.683 0.854 1.055 1.310 1.697 2.042 2.147 2.457 2.750 3.030 3.385 3.646
Critical Values for Normal Probability Plots 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
31 0.682 0.853 1.054 1.309 1.696 2.040 2.144 2.453 2.744 3.022 3.375 3.633
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
Sample Size, n Critical Value Sample Size, n Critical Value Sample Size, n Critical Value 32 0.682 0.853 1.054 1.309 1.694 2.037 2.141 2.449 2.738 3.015 3.365 3.622 Right tail Left tail Two tails
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 33 0.682 0.853 1.053 1.308 1.692 2.035 2.138 2.445 2.733 3.008 3.356 3.611 Area 5 12a
 5 0.880 13 0.932 21 0.952
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 34 0.682 0.852 1.052 1.307 1.691 2.032 2.136 2.441 2.728 3.002 3.348 3.601
 6 0.888 14 0.935 22 0.954 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 35 0.682 0.852 1.052 1.306 1.690 2.030 2.133 2.438 2.724 2.996 3.340 3.591
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
 7 0.898 15 0.939 23 0.956 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
36 0.681 0.852 1.052 1.306 1.688 2.028 2.131 2.434 2.719 2.990 3.333 3.582
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 37 0.681 0.851 1.051 1.305 1.687 2.026 2.129 2.431 2.715 2.985 3.326 3.574 a
 8 0.906 16 0.941 24 0.957 a a
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 38 0.681 0.851 1.051 1.304 1.686 2.024 2.127 2.429 2.712 2.980 3.319 3.566 a
 9 0.912 17 0.944 25 0.959 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2 2
39 0.681 0.851 1.050 1.304 1.685 2.023 2.125 2.426 2.708 2.976 3.313 3.558
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
10 0.918 18 0.946 30 0.960 40 0.681 0.851 1.050 1.303 1.684 2.021 2.123 2.423 2.704 2.971 3.307 3.551 Xa2 The area to the 2
2
X12a Xa2 The area to the right
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 X12a The area to the
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 50 0.679 0.849 1.047 1.299 1.676 2.009 2.109 2.403 2.678 2.937 3.261 3.496 right of this 2 2 of this value is a .
11 0.923 19 0.949 value is a. right of this 2
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 60 0.679 0.848 1.045 1.296 1.671 2.000 2.099 2.390 2.660 2.915 3.232 3.460 value is 12a. The area to the right
12 0.928 20 0.951 3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 70 0.678 0.847 1.044 1.294 1.667 1.994 2.093 2.381 2.648 2.899 3.211 3.435 a.
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998 of this value is 12
80 0.678 0.846 1.043 1.292 1.664 1.990 2.088 2.374 2.639 2.887 3.195 3.416 2
Source: S. W. Looney and T. R. Gulledge, Jr. “Use of the ­Correlation Coefficient with Normal Probability Plots,” ­American ­Statistician 39(Feb. 1985): 75–79.
90 0.677 0.846 1.042 1.291 1.662 1.987 2.084 2.368 2.632 2.878 3.183 3.402
Table  VII   Confidence Interval Critical Values, zA/2 Hypothesis Testing Critical Values
100 0.677 0.845 1.042 1.290 1.660 1.984 2.081 2.364 2.626 2.871 3.174 3.390
Level of Confidence Critical Value, zA/2 Level of Significance, Left-Tailed Right- Two- 1000 0.675 0.842 1.037 1.282 1.646 1.962 2.056 2.330 2.581 2.813 3.098 3.300
A Tailed Tailed z 0.674 0.842 1.036 1.282 1.645 1.960 2.054 2.326 2.576 2.807 3.090 3.291
0.90 or 90% 1.645
0.95 or 95% 1.96 0.10 - 1.28 1.28 { 1.645

0.98 or 98% 2.33 0.05 - 1.645 1.645 { 1.96

0.99 or 99% 2.575 0.01 - 2.33 2.33 { 2.575

Copyright © 2017 Pearson Education, Inc.

Z05_SULL3539_05_SE_Barrel.indd 2 10/27/15 7:10 PM

 Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­ Tables and Formulas for Sullivan, Statistics: Informed Decisions Using Data ©2017 Pearson Education, Inc­

Table I Table V Table VIII

Standard Normal Distribution
Random Numbers z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 Chi-Square (X2) Distribution
Column Number Area
Area in Area to the Right of Critical Value
Row 23.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002
right tail
Degrees of
23.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003
Number 01–05 06–10 11–15 16–20 21–25 26–30 31–35 36–40 41–45 46–50 z 23.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005 Freedom 0.995 0.99 0.975 0.95 0.90 0.10 0.05 0.025 0.01 0.005
23.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007
01 89392 23212 74483 36590 25956 36544 68518 40805 09980 00467 23.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010
t 1 — — 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7.879
02 61458 17639 96252 95649 73727 33912 72896 66218 52341 97141 22.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014 2 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.597
03 11452 74197 81962 48443 90360 26480 73231 37740 26628 44690 22.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019 3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838
04 27575 04429 31308 02241 01698 19191 18948 78871 36030 23980 22.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026 Table VII 4 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860
22.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
05 36829 59109 88976 46845 28329 47460 88944 08264 00843 84592 22.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048 t-Distribution 5 0.412 0.554 0.831 1.145 1.610 9.236 11.070 12.833 15.086 16.750
06 81902 93458 42161 26099 09419 89073 82849 09160 61845 40906 22.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064
Area in Right Tail 6 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548
07 59761 55212 33360 68751 86737 79743 85262 31887 37879 17525 22.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
22.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
7 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278
08 46827 25906 64708 20307 78423 15910 86548 08763 47050 18513 22.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143 df 0.25 0.20 0.15 0.10 0.05 0.025 0.02 0.01 0.005 0.0025 0.001 0.0005 8 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955
09 24040 66449 32353 83668 13874 86741 81312 54185 78824 00718 22.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183 9 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589
10 98144 96372 50277 15571 82261 66628 31457 00377 63423 55141 21.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233 1 1.000 1.376 1.963 3.078 6.314 12.706 15.894 31.821 63.657 127.321 318.309 636.619 10 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188
21.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294 2 0.816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 9.925 14.089 22.327 31.599
11 14228 17930 30118 00438 49666 65189 62869 31304 17117 71489 21.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367 11 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.757
12 55366 51057 90065 14791 62426 02957 85518 28822 30588 32798 21.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 3 0.765 0.978 1.250 1.638 2.353 3.182 3.482 4.541 5.841 7.453 10.215 12.924 12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.300
13 96101 30646 35526 90389 73634 79304 96635 06626 94683 16696 21.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559 4 0.741 0.941 1.190 1.533 2.132 2.776 2.999 3.747 4.604 5.598 7.173 8.610 13 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819
14 38152 55474 30153 26525 83647 31988 82182 98377 33802 80471 21.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681 5 0.727 0.920 1.156 1.476 2.015 2.571 2.757 3.365 4.032 4.773 5.893 6.869 14 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319
15 85007 18416 24661 95581 45868 15662 28906 36392 07617 50248 21.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 6 0.718 0.906 1.134 1.440 1.943 2.447 2.612 3.143 3.707 4.317 5.208 5.959
21.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
15 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801
16 85544 15890 80011 18160 33468 84106 40603 01315 74664 20553 21.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170 7 0.711 0.896 1.119 1.415 1.895 2.365 2.517 2.998 3.499 4.029 4.785 5.408 16 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267
17 10446 20699 98370 17684 16932 80449 92654 02084 19985 59321 21.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379 8 0.706 0.889 1.108 1.397 1.860 2.306 2.449 2.896 3.355 3.833 4.501 5.041 17 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718
18 67237 45509 17638 65115 29757 80705 82686 48565 72612 61760 20.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611 9 0.703 0.883 1.100 1.383 1.833 2.262 2.398 2.821 3.250 3.690 4.297 4.781 18 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156
19 23026 89817 05403 82209 30573 47501 00135 33955 50250 72592
20.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867 10 0.700 0.879 1.093 1.372 1.812 2.228 2.359 2.764 3.169 3.581 4.144 4.587 19 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582
20.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148
20 67411 58542 18678 46491 13219 84084 27783 34508 55158 78742 20.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451 11 0.697 0.876 1.088 1.363 1.796 2.201 2.328 2.718 3.106 3.497 4.025 4.437 20 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997
20.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776 12 0.695 0.873 1.083 1.356 1.782 2.179 2.303 2.681 3.055 3.428 3.930 4.318
21 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401
20.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121 13 0.694 0.870 1.079 1.350 1.771 2.160 2.282 2.650 3.012 3.372 3.852 4.221
Table II 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483 22 8.643 9.542 10.982 12.338 14.041 30.813 33.924 36.781 40.289 42.796
20.3 14 0.692 0.868 1.076 1.345 1.761 2.145 2.264 2.624 2.977 3.326 3.787 4.140
20.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859 23 9.260 10.196 11.689 13.091 14.848 32.007 35.172 38.076 41.638 44.181
Critical Values (CV) for Correlation Coefficient 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247 15 0.691 0.866 1.074 1.341 1.753 2.131 2.249 2.602 2.947 3.286 3.733 4.073
20.1 24 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.559
20.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641 16 0.690 0.865 1.071 1.337 1.746 2.120 2.235 2.583 2.921 3.252 3.686 4.015 25 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928
n CV n CV n CV n CV 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
17 0.689 0.863 1.069 1.333 1.740 2.110 2.224 2.567 2.898 3.222 3.646 3.965
3 0.997 10 0.632 17 0.482 24 0.404 18 0.688 0.862 1.067 1.330 1.734 2.101 2.214 2.552 2.878 3.197 3.610 3.922 26 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 19 0.688 0.861 1.066 1.328 1.729 2.093 2.205 2.539 2.861 3.174 3.579 3.883 27 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.195 46.963 49.645
4 0.950 11 0.602 18 0.468 25 0.396 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 28 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993
20 0.687 0.860 1.064 1.325 1.725 2.086 2.197 2.528 2.845 3.153 3.552 3.850
5 0.878 12 0.576 19 0.456 26 0.388 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 29 13.121 14.256 16.047 17.708 19.768 39.087 42.557 45.722 49.588 52.336
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 21 0.686 0.859 1.063 1.323 1.721 2.080 2.189 2.518 2.831 3.135 3.527 3.819 30 13.787 14.953 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672
6 0.811 13 0.553 20 0.444 27 0.381 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 22 0.686 0.858 1.061 1.321 1.717 2.074 2.183 2.508 2.819 3.119 3.505 3.792
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 23 0.685 0.858 1.060 1.319 1.714 2.069 2.177 2.500 2.807 3.104 3.485 3.768 40 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66.766
7 0.754 14 0.532 21 0.433 28 0.374 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
24 0.685 0.857 1.059 1.318 1.711 2.064 2.172 2.492 2.797 3.091 3.467 3.745 50 27.991 29.707 32.357 34.764 37.689 63.167 67.505 71.420 76.154 79.490
8 0.707 15 0.514 22 0.423 29 0.367 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 25 0.684 0.856 1.058 1.316 1.708 2.060 2.167 2.485 2.787 3.078 3.450 3.725 60 35.534 37.485 40.482 43.188 46.459 74.397 79.082 83.298 88.379 91.952
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 70 43.275 45.442 48.758 51.739 55.329 85.527 90.531 95.023 100.425 104.215
9 0.666 16 0.497 23 0.413 30 0.361 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 26 0.684 0.856 1.058 1.315 1.706 2.056 2.162 2.479 2.779 3.067 3.435 3.707
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 80 51.172 53.540 57.153 60.391 64.278 96.578 101.879 106.629 112.329 116.321
27 0.684 0.855 1.057 1.314 1.703 2.052 2.158 2.473 2.771 3.057 3.421 3.690
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 90 59.196 61.754 65.647 69.126 73.291 107.565 113.145 118.136 124.116 128.299
28 0.683 0.855 1.056 1.313 1.701 2.048 2.154 2.467 2.763 3.047 3.408 3.674
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 100 67.328 70.065 74.222 77.929 82.358 118.498 124.342 129.561 135.807 140.169
Table VI  1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
29 0.683 0.854 1.055 1.311 1.699 2.045 2.150 2.462 2.756 3.038 3.396 3.659
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 30 0.683 0.854 1.055 1.310 1.697 2.042 2.147 2.457 2.750 3.030 3.385 3.646
Critical Values for Normal Probability Plots 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
31 0.682 0.853 1.054 1.309 1.696 2.040 2.144 2.453 2.744 3.022 3.375 3.633
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
Sample Size, n Critical Value Sample Size, n Critical Value Sample Size, n Critical Value 32 0.682 0.853 1.054 1.309 1.694 2.037 2.141 2.449 2.738 3.015 3.365 3.622 Right tail Left tail Two tails
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 33 0.682 0.853 1.053 1.308 1.692 2.035 2.138 2.445 2.733 3.008 3.356 3.611 Area 5 12a
 5 0.880 13 0.932 21 0.952
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 34 0.682 0.852 1.052 1.307 1.691 2.032 2.136 2.441 2.728 3.002 3.348 3.601
 6 0.888 14 0.935 22 0.954 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 35 0.682 0.852 1.052 1.306 1.690 2.030 2.133 2.438 2.724 2.996 3.340 3.591
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
 7 0.898 15 0.939 23 0.956 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
36 0.681 0.852 1.052 1.306 1.688 2.028 2.131 2.434 2.719 2.990 3.333 3.582
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 37 0.681 0.851 1.051 1.305 1.687 2.026 2.129 2.431 2.715 2.985 3.326 3.574 a
 8 0.906 16 0.941 24 0.957 a a
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 38 0.681 0.851 1.051 1.304 1.686 2.024 2.127 2.429 2.712 2.980 3.319 3.566 a
 9 0.912 17 0.944 25 0.959 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2 2
39 0.681 0.851 1.050 1.304 1.685 2.023 2.125 2.426 2.708 2.976 3.313 3.558
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
10 0.918 18 0.946 30 0.960 40 0.681 0.851 1.050 1.303 1.684 2.021 2.123 2.423 2.704 2.971 3.307 3.551 Xa2 The area to the 2
2
X12a Xa2 The area to the right
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 X12a The area to the
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 50 0.679 0.849 1.047 1.299 1.676 2.009 2.109 2.403 2.678 2.937 3.261 3.496 right of this 2 2 of this value is a .
11 0.923 19 0.949 value is a. right of this 2
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 60 0.679 0.848 1.045 1.296 1.671 2.000 2.099 2.390 2.660 2.915 3.232 3.460 value is 12a. The area to the right
12 0.928 20 0.951 3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 70 0.678 0.847 1.044 1.294 1.667 1.994 2.093 2.381 2.648 2.899 3.211 3.435 a.
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998 of this value is 12
80 0.678 0.846 1.043 1.292 1.664 1.990 2.088 2.374 2.639 2.887 3.195 3.416 2
Source: S. W. Looney and T. R. Gulledge, Jr. “Use of the ­Correlation Coefficient with Normal Probability Plots,” ­American ­Statistician 39(Feb. 1985): 75–79.
90 0.677 0.846 1.042 1.291 1.662 1.987 2.084 2.368 2.632 2.878 3.183 3.402
Table  VII   Confidence Interval Critical Values, zA/2 Hypothesis Testing Critical Values
100 0.677 0.845 1.042 1.290 1.660 1.984 2.081 2.364 2.626 2.871 3.174 3.390
Level of Confidence Critical Value, zA/2 Level of Significance, Left-Tailed Right- Two- 1000 0.675 0.842 1.037 1.282 1.646 1.962 2.056 2.330 2.581 2.813 3.098 3.300
A Tailed Tailed z 0.674 0.842 1.036 1.282 1.645 1.960 2.054 2.326 2.576 2.807 3.090 3.291
0.90 or 90% 1.645
0.95 or 95% 1.96 0.10 - 1.28 1.28 { 1.645

0.98 or 98% 2.33 0.05 - 1.645 1.645 { 1.96

0.99 or 99% 2.575 0.01 - 2.33 2.33 { 2.575

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