Test Statistics on

Population Mean
Learner's Module in Statistics and
Probability
Quarter 4 ● Module 4 ● Week 4

WINNIE C. MARTES
Developer

Department of Education • Cordillera Administrative Region

NAME:________________________ GRADE AND SECTION ________________

TEACHER: ____________________ SCORE _____________________________
 Republic of the Philippines
DEPARTMENT OF EDUCATION
Cordillera Administrative Region
SCHOOLS DIVISION OF BAGUIO CITY
No. 82 Military Cut-off, Baguio City

Published by:
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Curriculum Implementation Division
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2021

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 What I Know
This pre-test will determine your prior knowledge on test statistics concerning
means. If you are able to answer all the test items correctly, then you may skip
studying this learning material and proceed to the next learning module.

Direction: Choose the letter of the correct answer and write it on each blank
provided. Use capital letters.

_____1. Which of the following test statistics is used when the population standard
deviation is known and ?
A. distribution B. distribution C. ̅ D.

_____2. What is the formula for distribution?

̅ ̅ ̅ ̅
A. B. C. D.
√ √

_____3. What is the formula for distribution?

̅ ̅ ̅ ̅
A. B. C. D.
√ √

_____4. Given: ̅ . What test statistic

must be used?
A. distribution B. distribution C. ̅ D.

_____5. What is the value of the test-statistic in number 4?

A. B. C. D.

_____6. Given: ̅ . What test statistic

must be used?
A. distribution B. distribution C. ̅ D.

_____7. What is the value of the test-statistic in number 6?

A. B. C. D.

_____8. Given: What

conclusion can be drawn from the given information?
A. Reject the null hypothesis. C.
B. Do not reject the null hypothesis. D.

______9. Given:
What conclusion can be drawn from the given information?
A. Reject the null hypothesis. C.
B. Do not reject the null hypothesis. D.

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For items 10-15, refer to this word problem: In one research, the average height
of Filipinos is 156.41 cm. Suppose a random sample of 25 Filipino citizens were
selected and it was determined that their mean height is 161.2 cm and the standard
deviation is 15 cm. Using 0.01 level of significance, can it be concluded that the
average height of Filipinos has increased? Assume normality over the population.

_____10. What is the null and alternative hypothesis in the problem above?
A. C.
B. D.

_____11. What is the level of significance?

A. B. C. D.

_____12. What test statistic should be used?

A. test B. test C. ̅ D.

_____13. Compute the test statistic.

A. B. C. D.

_____14. Determine the critical value.

A. B. C. D.

_____15. Based on the critical value and the computed value of the test statistic,
what conclusion can be drawn?
A. Reject . C. The critical value is low.
B. Do not reject . D.

What’s In

Activity: True or False

Direction: Write TRUE if the statement is correct and FALSE if incorrect. Write your
answers on the space provided.

_____1. The notations and are sample values.

_____2. The alternative hypothesis is a statement that there is no significant
difference between two given properties.
_____3. The hypotheses and show a one-tailed test since it
shows direction of the distribution.
_____4. The level of significance, denoted by (alpha), is related to the degree of
certainty required in order to reject the null hypothesis or not.

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_____5. A test statistic is a quantity calculated from the sample data and its value is
used to decide whether or not the null hypothesis should be rejected in a
hypothesis test.

What’s New
Consider this situation: The leader of the association of taxi drivers claimed that the
average daily take home pay of all taxi drivers in Baguio City is Php 400.00 due to
the current pandemic. A random sample of 100 taxi drivers were interviewed and it
was found out that their average daily take home pay is Php 425.00. Use a 0.05
level of significance to find out if the average daily take home pay of all taxi drivers in
Baguio City is different from Php 400.00. Assume that the population standard
deviation is Php 92.00. To test the claim of the leader of the association of taxi
drivers, what are the steps to be done?

The claim of the leader of the association of taxi drivers is considered a hypothesis.
So to help us determine whether the claim is acceptable or not, let us take a look at
the steps in hypothesis testing:

1 • State the null and alternative hypothesis.

2 • Determine the level of significance.

3 • Select the test statistic.

4 • Compute the test- statistic value.

5 • Determine the critical value.

6 • Draw a conclusion.

What Is It
The - test and the - test are two statistical tools used in hypothesis testing
concerning means. In this module, we will be studying tests for a single population
mean and difference of means tests.

Lesson 1: Test Statistics on Single Population Mean

The - test is the test statistic used in testing hypothesis when the population
standard deviation is known and the sample size is at least 30 ( ). In the
absence, however, of the population standard deviation, the sample standard
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deviation may be used. To find the - value using one sample mean, the formula
used is:
̅
where ̅ sample mean,
√
population mean,
sample size,
population standard deviation

AREAS UNDER THE NORMAL CURVE TABLE

The -test, on the other hand, is the test statistic used when the sample standard
deviation is known and the sample size is less than 30 ( ). To find the -
value using one sample mean, the formula used is:
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 ̅
with
√
where ̅ sample mean,
population mean,
sample size,
sample standard deviation
degrees of freedom

After computing the test- statistic value, decision-making and stating the conclusion
are done. Generally, if the absolute value of the computed value of or is greater
than the absolute value of the critical value (or tabular value) of or , the null
hypothesis is rejected. That is,
reject if | | | | for the -test and

reject if | | | | for the - test.

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Study the following examples:

Example 1: In the example given under What’s New, the leader of the association
of taxi drivers claimed that the average daily take home pay of all taxi
drivers in Baguio City is Php 400.00 due to the current pandemic. A
random sample of 100 taxi drivers was interviewed and it was found
out that their average daily take home pay is Php 425.00. Use a 0.05
level of significance to find out if the average daily take home pay of all
taxi drivers in Baguio City is different from Php 400.00. Assume that
the population standard deviation is Php 92.00.

Solution: Given: ̅

Step 1: State the null and alternative hypotheses.

(two-tailed test)

Step 2: Level of significance.

Step 3: Select the test statistic to be used. Since the population standard deviation
is known and , the appropriate test statistic to be used is the test:
̅
.
√

Step 4: Compute the test-statistic value.

̅

Step 5: Determine the critical value. The alternative hypothesis is non-directional,

hence, the two-tailed test shall be used. Divide by 2 then subtract the
quotient from 0.5000.

Use the Areas Under the Normal Curve Table. The area 0.4750 is under column
headed 0.06. Move along this row to the left until 1.9 under column headed is
reached. Therefore, At 5% level of significance, the critical values are

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 Rejection Rejection
region region
𝛼 𝛼
Non-rejection
region

𝜇
-1.96 -1.96
Critical Value Critical Value

Step 6: Draw a conclusion. Because the computed test-statistic value, , falls

within the rejection region or | | | | , reject the null
hypothesis. Conclude that the average daily take home pay of taxi drivers is
not equal to 400.00. This result is considered to be significant at
level.

Example 2: From a report in 2013, a typical Filipino drinks an average of 4.256

cups of coffee per week. Suppose a random sample of 12 senior
citizens were selected and it was determined that the mean and
standard deviation of their coffee consumption is 4.5 cups and 0.3989
cups, respectively. At 0.05 level of significance, does the sample data
suggest that there is a difference between the national average and
the sample mean from the senior citizens? Assume that the population
is approximately normal.

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Solution: Given: ̅

Step 1: State the null and alternative hypothesis.

(two-tailed test)

Step 2: Level of significance.

Step 3: Select the test statistic. Since and the population standard deviation
̅
is unknown, the appropriate test statistic is the test: with
√
.

Step 4: Compute the test-statistic value.

̅

Step 5: Determine the critical value.

a. Find the degrees of freedom.

b. Significance level:

Since the alternative hypothesis contains “ ”, we have a two-tailed test. This means
that the area of 0.05 will be divided into two tails. Use the Table of - Critical Values.
Locate 11 in the first column headed . Because the test is two-tailed with
, refer to the column headed 0.05 in two-tail. Therefore, At 5% level
of significance, the critical values are . We will reject if or
.

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 Rejection Rejection
region region
𝛼 𝛼
Non-rejection
region

𝜇
-2.201 -2.201
Critical Value Critical Value

Step 6: Draw a conclusion. Since the computed test-statistic value is not greater
than or | | | | , do not reject the null hypothesis.
Conclude that there is no sufficient evidence to indicate that the national
average of coffee consumption of Filipinos is different from the sample
mean from the senior citizens.

Example 3: According to a report, a Filipino household spends an average of Php

500.00 per day for food. Suppose you took random samples of 25
households and you determined how much each household spends for
food each day. The results revealed a mean of Php 490.00 and a
standard deviation of Php 21.50. Using 0.01 level of significance, can it
be concluded that the average amount spent per day for food of a
Filipino household has decreased? Assume normality over the
population.

Solution: Given: ̅

Step 1: State the null and alternative hypothesis.

(one-tailed test)

Step 2: Level of significance.

Step 3: Select the test statistic. Since population standard deviation is unknown, the
sample standard deviation is given, and which is less than 30, the
̅
appropriate test statistic is the test: with .
√

Step 4: Compute the test-statistic value.

̅

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Step 5: Determine the critical value.
a. Find the degrees of freedom.

b. Significance level:

The alternative hypothesis is directional. Hence, the one-tailed test shall be used.
Use the Table of - Critical Values. Locate 24 in the first column headed .
Because the test is one-tailed with , refer to the column headed 0.01 in
one-tail. The critical value is -2.492. There is a negative sign because of the
direction of the alternative hypothesis which is “<”.

Non-rejection
region
𝛼

𝜇
-2.492
Critical Value

Step 6: Draw a conclusion: Since the computed test- statistic value, ,

does not fall within the rejection region or | | | | , do not
reject the null hypothesis. Using a 0.01 level of significance, there is no
sufficient evidence to conclude that the average amount a Filipino
household spends per day for food has decreased.

Lesson 2: Tests for Difference Between Two Means

Experiments that compare two groups are common throughout science, including
social sciences, and industry. For example, we might want to compare the effects of
a new drug with traditional therapy, the fuel efficiency of two car engine designs, or
tolerance level to work-related stress between men and women.

In comparing two means, there are assumptions and conditions that need to be met
or checked. The first one is the independence assumption. The data in each group
must be drawn independently and at random from a homogeneous population, or
generated by a randomized comparative experiment. Second is the normal
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population assumption. We should check the assumption that the underlying
populations are each Normally distributed for both groups. And finally, the
independent groups assumption. The two groups we are comparing must be
independent of each other. Since no statistical test can verify this assumption, we
have to think of how the data were collected.

̅ ̅
The formula for test using two-sample means is
√

where : ̅ sample mean for group 1, ̅ sample mean for group 2

sample variance for group 1, sample variance for group 2
number of subjects in group 1, number of subjects in group 2

̅ ̅
The formula for test using two-sample means is
√ ( )

where : ̅ sample mean for group 1, ̅ sample mean for group 2

sample variance for group 1, sample variance for group 2
number of subjects in group 1, number of subjects in group 2

with degrees of freedom.

Pooled variance: where . The condition for the pooled test

for the difference between the means of two independent groups is that the
variances of the groups are assumed to be the same. That is, . Of course,
we can think about the standard deviations being equal instead.

Study the following examples:

Example 1: A bank is opening a new branch in one of two neighborhoods. One of

the factors considered by the bank is whether the average family
income (in thousand pesos) in the two neighborhoods differed. From
census records, the bank drew two random samples of 100 families
each and obtained the following information:

Neighborhood
Sample A Sample B

The bank wishes to test the null hypothesis that the two neighborhoods have equal
mean income. What should the bank conclude? Test the hypothesis using .

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Solution: For purposes of wider options and comparative discussions, solutions for
both one-tailed and two-tailed test are presented below.

1. One-tailed Test Using Two-Sample Means

Given: Sample A Sample B

a. State the null and alternative hypotheses.

b. Determine the significance level.
c. Determine the test statistic to be used. Since the variances and are
known and and are both equal to 100 ( ), use the - test for two-
sample mean:
̅ ̅

d. Compute the test-statistic value.

̅ ̅

e. Determine the critical value. The alternative hypothesis is directional so the

one-tailed test shall be used. That is, subtract from

Using the Areas Under the Normal Curve Table, the area 0.4500 is between 0.4495
and 0.4505. Therefore, At 5% level of
significance, the critical value is .

f. State the conclusion. Since | | | |, then | | | |. Therefore,

is rejected. The average family income of neighborhood A is higher than
neighborhood B at .

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2. Two-tailed Test Using Two-Sample Means
Given: Sample A Sample B

a. State the null and alternative hypotheses.

b. Determine the significance level.
c. Determine the test statistic to be used. Since variances and are known
and and are both equal to 100 ( >30), use the - test for two-sample
mean:
̅ ̅

d. Compute the test-statistic value.

̅ ̅

e. Determine the critical value. The alternative hypothesis is non-directional so

the two-tailed test shall be used. That is, divide by 2 then subtract
the quotient from 0.50.

Using the Areas Under the Normal Curve Table, the area 0.4750 is
Therefore, at 5% level of significance, the critical values are

f. State the conclusion. Since | | | | , then | | | |.

Therefore, is rejected. The two neighborhoods, A and B, do not have
equal average family income at .

Example 2: The weights of twelve babies who were given two types of diets were
recorded as follows:
Diet A Diet B

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At 0.01 level of significance, test whether the average weights of babies under diet
A is significantly higher than those under diet B. Assume that the date are normally
distributed.

Solution:
Given:
Diet A Diet B

a. State the null and alternative hypotheses.

b. Determine the significance level.

c. Determine the test statistic to be used. Since the variances and are
unknown and and are both equal to 12 ( 30), use the - test for two-
sample mean:
̅ ̅
with degrees of freedom.
√

Pooled variance: where

d. Compute the test-statistic value.

√ √
( )

e. Determine the critical value.

i. Find the degrees of freedom.

ii. Significance level:

The alternative hypothesis is directional so the one-tailed test shall be used. Using
the Table of - Critical Values. Locate 22 in the first column headed . Because the
test is one-tailed with , refer to the column headed 0.01 in one-tail. The
critical value is +2.508 because of the direction of the alternative hypothesis which
is “>”.

f. State the conclusion. Since | | | | , then | | | |.

Therefore, do not reject . Thus at , the average weight of the babies
under diet A is not significantly higher compared to the average weight of the
babies under diet B.

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 What’s More

Activity 1: Computing the test- statistic value

Direction: Choose the appropriate test statistic to be used based on the given
information then compute the test-statistic value.

1. Given: ̅

2. Given: ̅

Activity 2: Decision-making: Reject or Not!

Direction: Based on the given information, decide whether the null hypothesis should
be rejected or not. Write your answer on the blanks provided.

1. _________

2. _________

3. _________

4. _________

Activity 3: Hypothesis Testing

Direction: In the following problem, (a) state the null and alternative hypotheses, (b)
select and compute the test statistic, (c) determine the critical value and the rejection
region, and (d) draw a conclusion.

In a certain high school, male and female students were rated on their tolerance level
to school-related stress. The result are as follows:

Male students Female students

At 0.01 level of significance, test whether the tolerance level to school-related stress
of the male students is significantly higher than the female students. Assume that the
date is normally distributed.

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 What I Have Learned

To check your understanding on the test on population mean, complete the following
statements:

1. The steps in hypothesis testing are _____________________________________

_________________________________________________________________
_________________________________________________________________.

2. The is used when ____________________________________________

while the is used when ________________________________________.

What I Can Do

In a long bond paper, create an infographic chart about hypothesis testing concerning
means. Be creative!

Grading Rubric for Infographic Chart

Needs
Excellent Good Fair
Indicator Improvement
(5 pts) (4 pts) (3 pts)
(2 pts)
All information Information Most Few information
are detailed, are detailed, information are detailed,
accurate, accurate, are detailed, accurate,
Content relevant, and relevant, and accurate, relevant, and
properly cited. properly relevant, properly cited.
cited. and properly
cited.
Layout is Layout is Layout is Layout is
Infographic
aesthetically clear. generally somewhat
Design
pleasing. clear. unclear.
Additional Additional No Additional
elements such elements are additional elements are
as pictures are used but do elements used but there
Creativity incorporated not enhance are used. is no relevance
to enhance the to the content of
the infographic. the infographic.
infographic.

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 Assessment
This assessment aims to measure how much you have learned from this module.

Direction: Read and understand each statement before choosing the correct answer.
Write your answer on the space provided.

_____1. Which of the following test statistics is used when the population standard
deviation is known and ?
A. distribution B. distribution C. ̅ D.

_____2. What is the formula for distribution?

̅ ̅ ̅ ̅
A. B. C. D.
√ √

_____3. What is the formula for distribution?

̅ ̅ ̅ ̅
A. B. C. D.
√ √

_____4. Given: ̅ . What test statistic must be

used?
A. distribution B. distribution C. ̅ D.

_____5. What is the value of the test-statistic in number 4?

A. B. C. D.

_____6. Given: ̅ . What test statistic must be

used?
A. distribution B. distribution C. ̅ D.

_____7. What is the value of the test-statistic in number 6?

A. B. C. D.

_____8. Given: What

conclusion can be drawn from the given information?
A. Reject the null hypothesis. C.
B. Do not reject the null hypothesis. D.

______9. Given:
What conclusion can be drawn from the given information?
A. Reject the null hypothesis. C.
B. Do not reject the null hypothesis. D.

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For items 10-15, refer to this word problem: In one research, the average height
of Filipinos is 156.41 cm. Suppose a random sample of 25 Filipino citizens were
selected and it was determined that their mean height is 161.2 cm and the standard
deviation is 15 cm. Using 0.01 level of significance, can it be concluded that the
average height of Filipinos has increased? Assume normality over the population.

_____10. What is the null and alternative hypothesis in the problem above?
A. C.
B. D.

_____11. What is the level of significance?

A. B. C. D.

_____12. What test statistic should be used?

A. test B. test C. ̅ D.

_____13. Compute the test statistic.

A. B. C. D.

_____14. Determine the critical value.

A. B. C. D.

_____15. Based on the critical value and the computed value of the test statistic,
what conclusion can be drawn?
A. Reject . C. The critical value is low.
B. Do not reject . D.

Additional Activity

Activity: What do you think?

Answer each question in 1 – 3 sentences only.

1. When do you reject the null hypothesis?

_________________________________________________________________
_________________________________________________________________.

2. What conclusion can you derive if ? Explain

your answer.
_________________________________________________________________
_________________________________________________________________.

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 21
WHAT I KNOW/ ASSESSMENT WHAT I HAVE LEARNED: Answers may vary
1. B 11. C
2. B 12. A WHAT I CAN DO: Answers vary
3. C 13. D
4. B 14. B
5. D 15. B
6. A
7. C
8. A
9. B
10. D
WHAT’S MORE ADDITIONAL ACTIVITY: Answers vary
Activity 1:
1. 𝑧 𝑡𝑒𝑠𝑡; 𝑧
2. 𝑡 𝑡𝑒𝑠𝑡 𝑡
Activity 2: Activity 3:
1. Reject 𝐻𝑜 a. 𝐻 𝜇 𝜇 𝐻𝑎 𝜇 𝜇
2. Do not reject 𝐻𝑜 b. 𝑡 𝑡𝑒𝑠𝑡 𝑡
3. Do not reject 𝐻𝑜 c. critical value:
4. Reject 𝐻𝑜 d. Reject 𝐻𝑜
ANSWER KEY
 REFERENCES

Books:

Bock, David et al. 2007. Stats Modeling the World, 473-565. Teacher’s Edition

Mercado, Jesus and Orines, Fernando. 2016. Statistics and Probability for Senior High
School, 25-41. Phoenix Publishing House, Inc.

Shio, Christian Paul and Reyes, Maria Angeli. 2017. Statistics and Probability for Senior High
School, 223-232. C & E Publishing, Inc.

Zorilla, Roland et al. 2016. Statistics and Probability for Senior High School, 84-97. Mutya
Publishing House, Inc.

Lubrica, Maria Azucena B. Statistics for the Social Sciences, 38-49. Department of Math-
Physics-Statistics, College of Arts and Sciences, Benguet State University

Online sources:

https://www.youtube.com/watch?v=5ABpqVSx33I Accessed: April 19, 2021

https://www.youtube.com/watch?v=OaiLoJdsEyc Accessed: April 23, 2021

https://www.youtube.com/watch?v=NkGvw18zlGQ Accessed: April 23, 2021

https://www.youtube.com/watch?v=7vcu8wjWO6w Accessed: April 30, 2021

https://www.youtube.com/watch?v=zfeY94knNNs Accessed: April 30, 2021

https://www.youtube.com/watch?v=jxadMxJj8sk Accessed: April 30, 2021

https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf

https://onlinelibrary.wiley.com/doi/pdf/10.1002/0471308889.app2

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For inquiries or feedback, please write or call:
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No. 82 Military Cut-off Road, Baguio City
Telefax: 442-4326 / 422-7819
Email Address: depedbaguiocity@gmail.com
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