StatProba Math Grade11 Q4 2022-2023
StatProba Math Grade11 Q4 2022-2023
StatProba Math Grade11 Q4 2022-2023
STATISTICS &
PROBABILITY
QUARTER 4
Module 4
ii
DO_Q4_STATISTICS & PROBABILITY_GRADE 11_MODULE4
RESOURCE TITLE: Statistics & Probability
Alternative Delivery Mode
Quarter 4 – Week 1-10
Revised Edition, 2023
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11
STATISTICS &
PROBABILITY
QUARTER 4
Module 4
ii
Introductory Message
For the facilitator:
Welcome to the Statistics and Probability for Grade 11 Alternative Delivery Mode (ADM)
Module on Understanding Hypothesis Testing!
This module was collaboratively designed, developed and reviewed by educators both from
public and private institutions to assist you, the teacher or facilitator in helping the learners
meet the standards set by the K to 12 Curriculum while overcoming their personal, social,
and economic constraints in schooling.
This learning resource hopes to engage the learners into guided and independent learning
activities at their own pace and time. Furthermore, this also aims to help learners acquire
the needed 21st century skills while taking into consideration their needs and circumstances.
In addition to the material in the main text, you will also see this box in the body of the
module:
As a facilitator you are expected to orient the learners on how to use this module. You also
need to keep track of the learners' progress while allowing them to manage their own learning.
Furthermore, you are expected to encourage and assist the learners as they do the tasks
included in the module.
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For the learner:
Welcome to the Statistics and Probability for Grade 11 Alternative Delivery Mode (ADM)
Module on Understanding Hypothesis Testing!
The hand is one of the most symbolized part of the human body. It is often used to depict
skill, action and purpose. Through our hands we may learn, create and accomplish. Hence,
the hand in this learning resource signifies that you as a learner is capable and empowered
to successfully achieve the relevant competencies and skills at your own pace and time. Your
academic success lies in your own hands!
This module was designed to provide you with fun and meaningful opportunities for guided
and independent learning at your own pace and time. You will be enabled to process the
contents of the learning resource while being an active learner.
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In this portion, another activity will be given to
Additional Activities
you to enrich your knowledge or skill of the lesson
learned. This also tends retention of learned
concepts.
This contains answers to all activities in the
Answer Key
module.
1. Use the module with care. Do not put unnecessary mark/s on any part of the module.
Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer What I Know before moving on to the other activities included
in the module.
3. Read the instruction carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through with it.
If you encounter any difficulty in answering the tasks in this module, do not hesitate to
consult your teacher or facilitator. Always bear in mind that you are not alone.
We hope that through this material, you will experience meaningful learning and gain
deep understanding of the relevant competencies. You can do it!
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11
STATISTICS &
PROBABILITY
Quarter 4-Module 4
Lesson 1
Understanding Hypothesis
Testing
vi
This module was designed and written with you in mind. It is here to help you master
the sampling and sampling distributions. The scope of this module permits it to be
used in many different learning situations. The language used recognizes the diverse
vocabulary level of students. The lessons are arranged to follow the standard
sequence of the course. But the order in which you read them can be changed to
correspond with the textbook you are now using.
Let’s see how much you already know about this lesson.
Determine whether the statement is True or False. If False, modify the underlined
word/s to make it true.
_____________1.) The area under the normal curve is 1.
_____________2.) Under the normal curve, there are many z-values.
_____________3.) The level of significance, a = 0.01 gives 99% accuracy.
_____________4.) The level of significance, a = 0.05 gives 0.95% accuracy.
_____________5.) In a given problem, the notations µ and 𝝈 are sample values.
From the activity above, you have decided whether the statement is true or false.
In decision making, you usually follow certain processes: collect evidences, weigh
alternatives and decide.
In Statistics, decision making starts with a concern about a population
regarding its characteristics denoted by parameter values. We might be interested in
the population parameter like the mean and the proportion. For example, a
fisherman looks into several factors before deciding to go out to catch fish in the sea.
In the same manner, a farmer’s decision on when to plant his crops, and a politician
in a community decide to approve an agenda on environmental awareness are some
examples that can be addressed in procedures in Statistics called hypothesis testing.
Hypothesis Testing is another area in Inferential Statistics. It is a decision-
making process for evaluating claims about a population based on the characteristics
of a sample purportedly coming from that population. The decision is whether the
characteristic is acceptable or not.
There are two types of hypotheses: the null hypothesis and alternative
hypothesis.
Null hypothesis is the hypothesis to be tested. It states an exact value about the
parameter. When the null hypothesis is rejected, this leads to another option, which
is the alternative hypothesis that allows for the possibility of many values.
Level of Significance
The next step in hypothesis testing after the statement of the hypotheses is
the setting of the standard or criterion on which the decision will be based.
Apparently, there are only two possible decisions to make in the process of
hypothesis testing- either “reject Ho” (accept Ha) or “do not reject Ho” (reject Ha). This
decision to reject the null hypothesis is called significance and it should be based
on a set of criteria of judgment called the level of significance, denoted using the
Greek lower-case alpha, α.
➢ Significance is reached when the p-value of the statistic is less than the level
of significance.
➢ In general, statisticians arbitrarily set the commonly used levels of
significance,
at 1%, 5% and 10%.
Example 4:
It has been established that a particular teaching strategy improves math
performance. However, the p-value taken from your experiment at an alpha-value of
0.05 was 0.15. Thus, you did not reject the null hypothesis and concluded that there
is no significance between the strategy and math performance. What type of decision
is illustrated in this example?
Answer: This illustrates Type II error because there is really significance in the
population between the teaching strategy and math performance, but you
did not find any significance in your sample.
Probability of Committing a Type I and Type II error
In decisions that we make, we form conclusions and these conclusions are the
bases of our actions. But this is not always the case in Statistics because we make
decisions based on a sample information. The best way we can do is to control the
probability with which an error occurs.
The probability of committing a Type I error is denoted by Greek letter α (alpha)
while the probability of committing a Type II error is denoted by β (beta)
The following table shows the probability with which decisions occur.
Table 2. Types of Errors
Error in Decision Type Probability Correct Type Probability
Decision
Reject a true Ho I Α Accept a true Ho A 1-α
Accept a false Ho II Β Reject a false Ho B 1-β
Parameter
Parameter is defined as any numerical quantity that characterizes a given population
or some of its aspects. It means that, the parameter tells us something about the
whole population.
However, the numerical measure that is calculated from the sample is
called statistic. Statistic is a known number and a variable that depends on the
portion of the population.
A parameter denotes the true value that would be obtained if a census
rather than a sample was undertaken.
Directions: State the null and the alternative hypotheses of the following
statements.
1. A medical trial is conducted to test whether or not a new releases medicine
reduces uric acid by 40%.
: ____________________________________________________
: ____________________________________________________
2. Supposed, we want to test whether the general average of students in Math is
different from 82%.
: ____________________________________________________
: ____________________________________________________
3. We want to test whether the mean height of Grade 7 students is 56
inches.
: ____________________________________________________
: ____________________________________________________
4. We want to test if BNHS students take more than four years to graduate from
high school, on the average.
: ____________________________________________________
: ____________________________________________________
5. We want to test if it takes less than 45 minutes to answer the summative test in
Mathematics.
: ____________________________________________________
: ____________________________________________________
Directions: Determine if one-tailed test or two-tailed test fits the given alternative
hypothesis.
1. The enrolment in junior high schools is not the same as the enrolment in the
senior high schools.
2. The standard deviation of their height is not equal to 7 inches.
3. The average number of internet users this year is significantly higher as
compared last year.
4. Male Grade 8 and Grade 11 students differ in height on average.
5. Miya’s grade is higher compared to her previous grade.
1. What kind of parameter is applied in the given situation? “The mean height of all
Grade 10 students is 170 cm.”
A. mean B. variance C. proportion D. standard deviation
2. A licensed teacher claims that more than 40 % of all education graduates passed
the licensure examination for teachers. What kind of parameter is used in this
claim?
Anna wants to estimate the average shower time of teenagers. From the sample of 50
teenagers, she found out that it takes 5 minutes for teenagers to shower
Give 5 situation in your life that you experienced: (3) Correct decision (1) Type I
error and (1) Type II error.
Belecina, R., Baccay, E., & Mateo, E. (2016). Statistics and Probability. Manila,Philippines:
REX Book Store Inc.
Bluman, A. (2018). Elementary Statistics: A Step by Step Approach 10th edition. McGraw
Hill. New York, USA.
Canlapan, R. (2016). Statistics and Probability. Makati, Philippines: Diwa Learning System
Inc.
Keller, Warrack. (2003). Statistics For Management and Economics. California USA:
Thomson Learning, Inc.
Levine, et. al. (2005). Statistics: A Handbook for Managers. New Jersey: Prentice Hall.
PERCDC Learnhub
Walpole, R., Myers, R., Myers, S., and Ye, K., (2012). Probability and Statistics for Engineers
and Scientists 9th edition. Pearson Education Inc. Massachusetts, USA.
3. The cholesterol levels in a certain population have mean of 200 and standard
deviation 20. The cholesterol levels for a random sample of 9 individuals are
measured and the sample mean x is determined. What is the z-score for a sample
mean x=180?
In statistics, hypothesis testing is a way for you to test the results of a survey or
experiment to see if you have meaningful results. You're basically testing whether
your results are valid by figuring out the odds that your results have happened by
chance. In addition, it allows you to collect samples and make decision based on
facts, not on how you feel or what you think is right. To be able to prove your
assumptions, you must state first the null and alternative hypotheses.
Direction: Identify the situations which illustrate inequalities. Then write the
inequality model in the appropriate column.
Hypothesis
H0: The exposure to sunlight Ha: The exposure to sunlight
does not affect the growth of does affect the growth of the
the plant. plant.
Example:
1. A study was conducted to look at the average time students exercise. A
researcher claimed that in average, students exercise less than 12 hours per
month. In a random sample size n=110, it was found that the mean time
students exercise is x̄ = 11.3 hours per month with s = 6.40 hours per month.
Since n=110, the sample size is large and variance is unknown. Hence, z-
test is the appropriate tool. (Central Limit Theorem)
2. An English teacher wanted to test whether the mean reading speed of
students is 540 words per minute. A sample of 10 students revealed a
sample mean of 520 words per minute with a standard deviation of 5 words
per minute. At 0.05 significance level, is the reading speed different from 540
words per minute?
The sample size (n) is 10 which is less than 30 and sample standard
deviation (5 words per minute) was given. Therefore, the appropriate test
is t-test.
Directions: Identify the appropriate test statistic to be used in each problem. Write
z-test or t-test on a separate sheet of paper.
___________1. A sample of n=20 is selected from a normal population, mean = 53
and s= 10.
___________3. Knowing from a previous study that the average of athletes is 60, an
athletic adviser asked how his soccer players are academically doing as compared to
other student athletes. After an initiative to help improve the average of student
athletes, the adviser randomly selected 15 soccer players and found 80 as the average
with standard deviation of 1.20.
___________5. It was known that the number of tickets purchased by students at the
ticket window for the volleyball match of two popular universities followed a
distribution that has mean of 500 and standard deviation of 8.7. Suppose that a few
hours before the start of one of these matches, there are 100 eager students standing
in line to purchase tickets. If there are 250 tickets remaining, what is the probability
that all 100 students will be able to purchase the tickets they want?
Belecina, R., Baccay, E., & Mateo, E. (2016). Statistics and Probability. Manila,Philippines:
REX Book Store Inc.
Bluman, A. (2018). Elementary Statistics: A Step by Step Approach 10th edition. McGraw
Hill. New York, USA.
Canlapan, R. (2016). Statistics and Probability. Makati, Philippines: Diwa Learning System
Inc.
Keller, Warrack. (2003). Statistics For Management and Economics. California USA:
Thomson Learning, Inc.
Levine, et. al. (2005). Statistics: A Handbook for Managers. New Jersey: Prentice Hall.
PERCDC Learnhub
Walpole, R., Myers, R., Myers, S., and Ye, K., (2012). Probability and Statistics for Engineers
and Scientists 9th edition. Pearson Education Inc. Massachusetts, USA.
STATISTICS &
PROBABILITY
Quarter 4-Module 4
Lesson 3
The Rejection Region and
Critical Values
Directions: Read and analyze each item carefully, then circle the letter of the
correct answer from the given choices.
In this lesson, the learners will understand the concept of rejection region and critical
values. Ideas about types of error will also be presented.
This goes to show that critical values play an important role in establishing
the region/s under the curve where the hypothesis being tested may be rejected or
not. The region in which the hypothesis must be rejected is called the rejection
region.
Aside from establishing the hypothesis and identifying the appropriate test
statistic, there are other elements of hypothesis testing relevant to decision making.
As we can naturally commit mistakes, one of these relevant elements is the concept
of error. It was mentioned above that there are two possible decisions. Also, the null
hypothesis may either be true or false. Hence, there are 4 possible combinations of
decisions and truth values of the null hypothesis.
Interestingly, only two of these four outcomes are correct. The other two are
errors. These errors are named as Type I error and Type II error. Study the diagram
given below.
Reject Ho Do not reject Ho
TYPE I CORRECT
Ho is true
ERROR DECISION
We can note from the diagram that a Type I error is committed when a true
hypothesis is rejected while a Type II error is committed when you fail to reject a false
hypothesis.
How do these errors relate in real life? Let us see the illustrations below.
Illustration 1: A man who insists that he stands 5’10” when in fact, his height is only
5’8”. In this situation, the man is said to commit a Type I error since he is rejecting
the idea that he just stands 5’8”.
The figure on the left shows the rejection regions under the
normal curve for a non-directional (two-tailed) test. This time,
notice that the entire area under the curve is divided into three
parts by the critical values. The rejection regions are seen on
both tails, which means that 𝛼 has been equally distributed.
Remember that these regions serve us our guide in decision making. If the
computed test statistic falls in the rejection region, then we must reject H o. If the test
statistic falls outside the rejection region, then we do not reject Ho.
As a remark, the curve to be used is based on whether the population variance
is known or not.
Sample Problem:
1. Assuming that the population standard deviation is known, sketch the rejection
region for a two-tailed test with 95% confidence. Does z = 1.68 fall in the rejection
region?
Solution:
Since the population standard deviation is assumed to be known, we will use
the z-distribution (normal distribution). Also, the 95% confidence level implies that
𝛼 = 0.05. Further, the two-tailed test implies that we must consider 𝛼/2 = 0.025
since the probability is distributed on both tails. Thus, the critical value is the
corresponding z-value for 1 – 0.025 = 0.975. Using the z-table, we find that the
critical values are -1.96 and 1.96. The sketch of the rejection region is shown
below.
3. A random sample of 250 bottles of juice drink were taken and was found to have an average content
that is less than the company’s claim that each bottle contains 500 mL of juice drink. Suppose that
an appropriate test statistic revealed a value of -1.75 at 95% confidence, sketch the rejection region
and locate test statistic value.
Solution:
It is seen from the problem that the population standard deviation is unknown but with the
sample size of 250, which is large enough, we can make use of the Central Limit Theorem and
consider the z-distribution. With 95% confidence, it shows that 𝛼 = 0.05. Thus, 1 − 𝛼 = 1 −
0.05 = 0.95.
Further, the phrase ‘less than’ indicates that we have a one-tailed test. Thus, we verify now if z
= -1.75 lies on the rejection region or not. The sketch is shown below.
A. Decide whether each statement is TRUE or FALSE. Write T for True and F
for False.
_____1. We use t-distribution when the population standard deviation is known.
_____2. In a one-tailed test, the rejection region is found on both tails of a
distribution.
_____3. The critical values divide the curve into rejection and non-rejection
regions.
_____4. Type II error is committed when a false hypothesis is not rejected.
_____5. The probability of not committing a Type I error is 1 − 𝛼.
1. Locate z = 1.96 under the curve and sketch the rejection region for a one-tailed
test with 99% confidence. Is z found on the rejection region?
2. One hundred packs of potato chips were selected to verify the manufacturer’s
claim that the mean weight of each pack is 36 grams. At 95% confidence, are the
mean weights of the sample and the population significantly different after
knowing that the test statistic is 1.83? Sketch the rejection region and locate the
test statistic.
3. Previous records of a supermarket revealed that their goers have an average
budget of Php 1, 550. Suppose the grocery budgets of 20 randomly chosen
supermarket goers were taken and a test statistic value of 2.15 was computed.
Is there enough evidence to say that there is no significant difference between
the sample mean and the population mean at 95% confidence? Locate the test
statistic and sketch the rejection region.
Belecina, R., Baccay, E., and Mateo E. (2016). Statistics and Probability. Rex
Publishing House. Manila.
Bluman, A. (2018). Elementary Statistics: A Step by Step Approach 10 th edition.
McGraw Hill. New York, USA.
https://www.khanacademy.org/math/statistics-probability/significance-tests-one-
sample/idea-of-significance-tests/v/simple-hypothesis-testing
https://www.khanacademy.org/math/statistics-probability/significance-tests-one-
sample/more-significance-testing-videos/v/hypothesis-testing-and-p-values
https://www.statisticshowto.com/probability-and-statistics/hypothesis-testing/
STATISTICS &
PROBABILITY
Quarter 4-Module 4
Lesson 4
The Test Statistic
Directions: Read and analyze each item carefully and circle the letter of the
correct answer from the given choices.
***Refer to the statements below to answer item numbers 1 and 2.
I. The distribution is normal or approximately normal.
II. The population standard deviation is known.
III. The population standard deviation is unknown.
IV. The sample size is greater than or equal to 30.
1. Which of the conditions above must be met so that one can compute for the z-
test statistic?
A. I and II B. I and III C. I, II and IV D. I, III and IV
2. Which of the conditions above must be met so that one can compute for the t-
test statistic?
A. I and II B. I and III C. I, II and IV D. I, III and IV
3. Which values must be compared in order to make a decision about the null
hypothesis?
A. test statistic and confidence level
B. test statistic and critical value
C. test statistic and degrees of freedom
D. test statistic and significance level
4. Given that 𝑛 = 135; 𝑋̅ = 15; 𝜇 = 8; 𝜎 = 1.23, what test statistic is appropriate to
use?
A. t-test statistic C. either A or B
B. z-test statistic D. insufficient data
̅
5. Given the following: 𝑛 = 75; 𝑋 = 6.9; 𝜇 = 6.4; 𝑠 = 1.5, what must be the value of the
appropriate test statistic?
A. z = -2.89 B. z = 2.89 C. t = -2.89 D. t = 2.89
***Refer to the situation below to answer item numbers 6 to 10.
A pool of researchers claims that the average age of schooling among children
in a certain district is 4.8 years with a standard deviation of 0.21. A pre-school
teacher attempted to verify this claim by taking the ages of 360 first-time
schoolers in the said district and found out that the average age is 4.12 years.
6. What test statistic must be computed to test the claim of the researchers?
A. t-test statistic C. either A or B
B. z-test statistic D. insufficient data
7. What is the correct value of the test statistic?
A. -61.44 B. -58.42 C. -0.48 D.-0.02
8. What must be the appropriate critical value if 99% significance level was used?
In the previous lesson, we learned about critical values and their role in
establishing the rejection region. To complete the scenario towards arriving with a
certain decision on whether to reject the null hypothesis or not, one must be able to
compute for the test statistic accurately.
The calculation of the test statistic mainly depends on whether or not the
population standard deviation is known as well as on the size of the sample. In this
lesson, we introduce the formula and the procedures for computing an appropriate
test statistic which will be used to arrive at a correct decision that may eventually
lead to sound conclusions.
Sample Problem:
1) Previous records revealed that the mean salary of the high school teachers in a
municipality is Php 16, 250 with a standard deviation of Php 1, 400. A sample of
50 teachers were taken and was reported to have a mean salary of Php 18,000. At
95% confidence level, do we have enough evidence to believe what the records
revealed?
Solution:
Since the sample size is 50, which is greater than or equal to 30 and that
the population standard deviation is known, we compute for the z-test statistic.
Substituting the known values to our formula for z-test, we obtain the following:
At, 95% confidence interval, the critical value for z is 1.96. Clearly, the
computed z-test statistic which is 8.839 is greater than the z-critical value of 1.96.
Thus, we reject the null hypothesis stating that the population mean salary and
the sample mean salary are statistically equal.
Therefore, the sample mean is statistically different from that of the population
mean. This implies that the selected high school teachers have significantly
different salary as compared to the population and so, there is no enough evidence
to believe what the records have revealed.
2) A medical report claims that the number of infections per week at a certain
hospital in a province is 12.7. A random sample of 9 weeks had a mean number
of 11.4 infections with a standard deviation of 0.6. Is there enough evidence to
support the claim at 95% confidence level?
Solution:
Given that the sample size is 9, which is less than 30 and that the population
standard deviation is unknown, we compute for the t-test statistic. Applying the
formula for this test value, we have the following:
𝑋̅ − 𝜇
𝑡=
𝑠/√𝑛
11.4 − 12.7
𝑡=
0.6/√9
42 DO_Q4_STATISTICS & PROBABILITY_GRADE 11_MODULE4
−1.3
𝑡= = −6.5
0.2
At 95% confidence level, 𝛼 = 0.05. Also, when 𝑛 = 9, 𝑑𝑓 = 9 − 1 = 8. Using the
t-table, these lead to the t-critical value of ±2.306. Comparing the absolute values,
we can say that the computed test statistic is greater than the critical value.
Hence, we reject the null hypothesis stating that the population mean, and the
sample mean are statistically equal.
Therefore, the sample mean is statistically different from that of the
population mean. This implies that the selected weeks have significantly different
number of infections as compared to the population and so, there is no enough
evidence to support what the medical report claims.
• To compute for a test statistic, we subtract the expected value form the
observed value and divide the result by the standard error.
• We use z-test when 𝑛 ≥ 30 or when the population is normally distributed and
𝑋̅ −𝜇
𝜎 is known. The formula for the z-test statistic is given by 𝑧 = .
𝜎/√𝑛
• We use t-test when the population is normal or approximately normal and 𝜎
𝑋̅ −𝜇
is unknown. The formula for the t-test statistic is given by 𝑡 =
𝑠/√𝑛
• If the absolute value of computed test statistic is greater than or equal to the
critical value, the null hypothesis is rejected and if the absolute value of
computed test statistic is less than the critical value, we do not reject the null
hypothesis.
• Conclusions are inferences that we can draw from the context of the situation
based on whether we have rejected the null hypothesis or not.
Directions: Decide whether each statement is TRUE or FALSE. Write T for True
and F for False.
_______1. We use z-test statistic when the population standard deviation is
unknown.
_______2. The inferences that we can draw out of a decision are called
conclusions.
Directions: Read and analyze each item carefully and circle the letter of the
correct answer from the given choices.
***Refer to the statements below to answer item numbers 1 and 2.
I. The distribution is normal or approximately normal.
II. The population standard deviation is known.
III. The population standard deviation is unknown.
IV. The sample size is greater than or equal to 30.
1. Which of the conditions above must be met so that one can compute for the t-
test statistic?
A. I and II B. I and III C. I, II and IV D. I, III and IV
2. Which of the conditions above must be met so that one can compute for the z-
test statistic?
A. I and II B. I and III C. I, II and IV D. I, III and IV
3. Which values must be compared to decide about the null hypothesis?
A. test statistic and critical value
B. test statistic and significance level
C. test statistic and confidence level
D. test statistic and degrees of freedom
4. Given that 𝑛 = 50; 𝑋̅ = 12; 𝜇 = 11; 𝜎 = 0.12, what test statistic is appropriate to
use?
A. t-test statistic C. either A or B
B. z-test statistic D. insufficient data
̅
5. Given the following: 𝑛 = 25; 𝑋 = 6.9; 𝜇 = 6.4; 𝑠 = 1.5, what must be the value of the
appropriate test statistic?
A. z = -1.667 B. z = 1.667 C. t = -1.667 D. t = 1.667
***Refer to the situation below to answer item numbers 6 to 10.
A pool of researchers claims that the average age of schooling among children
in a certain district is 4.8 years with a standard deviation of 0.21. A pre-school
teacher attempted to verify this claim by taking the ages of 20 first-time schoolers
in the said district and found out that the average age is 4.12 years.
6. In testing the claim of the researchers, what test statistic must be determined?
A. t-test statistic C. either A or B
B. z-test statistic D. insufficient data
7. Which of the following is the correct value of the test statistic?
A. 14.481 B. 10.481 C. -0.032 D. -14.481
8. Suppose that 95% confidence level was used, what must be the appropriate
critical value?
A. ±1.65 B. ±1.96 C. ±2.093 D. ±2.064
9. How do the absolute values of the test statistic (TS) and the critical value (CV)
compare?
A. TS = CV C. TS < CV
B. TS > CV D. insufficient data
Additional Activities
Directions: Read and analyze each item carefully, then circle the letter of the
correct answer from the given choices.
In the past lessons, you learned how to formulate null and alternative
hypotheses for population mean, identify critical values, compute test statistic,
decide, and state conclusions based on the results. The test of hypothesis concerning
population mean may be described as a decision-making process about a certain
claim concerning a population.
The previous lessons have indeed allowed you to experience how this entire
process is performed in a piece-by-piece approach. In this lesson, we will look at all
these procedures as a one whole big picture. The test of hypothesis may be conducted
in three ways namely – traditional method, p-value method, and confidence interval
method. For this lesson we will only tackle about the traditional method and its steps
are summarized below.
Steps in Hypothesis Testing using the Traditional Method
1. State the hypothesis and identify the claim. 4. Make a decision.
2. Find the critical value. 5. State the conclusion.
3. Compute for the appropriate test statistic.
Sample Problem 1:
According to the latest data published by the World Health Organization
(WHO) in 2018, the life expectancy of male Filipinos is 66.2 years. A random sample
of 50 recorded deaths among male Filipinos was taken and was found to have a mean
of 64.6 years. Assuming that the population standard deviation is 7.2 years, does
this seem to indicate that the mean life span of male Filipinos is less than 66.2 years?
Use 0.05 level of significance.
Steps Solution
1. State the hypothesis The following are the hypotheses:
and identify the claim. Ho: 𝜇 = 66.2 H1: 𝜇 < 66.2 (claim)
2. Find the critical value. Since the population standard deviation is known, then
the appropriate distribution is z. Also, the hypotheses
are directional and so it implies a one-tailed test with 𝛼 =
0.05.
Using the z-table, the critical z-value is -1.65.
Solution:
In the above problem, we formulate the following hypotheses:
Ho: 𝑝 = 0.67 (claim) H1: 𝑝 ≠ 0.67
b. A survey revealed that more than 46% of working professionals dine-in at fast
food stores daily. A pool of researchers tested this survey result by taking 75
working professionals with 52 of them agreeing the result.
Solution:
In the above problem, we formulate the following hypotheses:
Ho: 𝑝 = 0.46 H1: 𝑝 > 0.46 (claim)
A recent survey of 200 people revealed that the mean time spent in watching
television of teenagers is 4.2 hours. Previous national records say that the mean time
was 3.8 hours with standard deviation of 0.3 hours. Do the survey results
significantly differ from previous records at 0.05 level of significance?
• There are five steps in the Traditional Method of Hypothesis Testing. These
are (1) State the hypothesis and identify the claim; (2) Find the critical
53 DO_Q4_STATISTICS & PROBABILITY_GRADE 11_MODULE4
value; (3) Compute for the appropriate test statistic; (4) Make a decision;
and (5) State the conclusion.
• Situations in which a percentage of the population is given instead of
means involve population proportion.
• When the Central Limit Theorem is used, the appropriate form of the test
statistic is z.
A. State the null and alternative hypotheses for each of the following.
1. A recent report indicated that 72% of teachers spend more than 8 hours in
doing schoolwork in this current work-from-home arrangement. To verify
this claim, a researcher took 80 teachers, and revealed that 63 of them
affirms the report.
2. A survey revealed that less than 36% of children ages 8 to 10 years old are
exposed to computer games daily. A pool of researchers tested this survey
result by taking 105 children on the given age bracket with 41 of them
agreeing the result.
B. Read and analyze the problem below then solve.
A random sample of 150 bottles of juice drink were taken and was found to
have an average content of 318 mL with a standard deviation of 2.2 mL. This
average content is less than the company’s claim that each bottle contains 330
mL of juice drink. At 0.05 significance level, do the contents of the juice drink of
the sample significantly differ to that of the population?
Directions: Read and analyze each item carefully, then circle the letter of the
correct answer from the given choices.
The ABM Coffee Company claims that 20% of the coffee drinkers in Pinalagad,
Malinta prefer their brand, the Ang Barako Mo coffee. To test the claim, a group of CNHS
Grade 11 students conducted a survey and this is what they found out. Out of the 500
randomly selected residents, 95 indicated that ABM coffee is the reason why they wake up
in the morning.
Can we believe the claim? Is the claim true?
If the favorable responses were 50 or 60, it seems very reasonable to reject the
claim. On the other hand, if the number of people who say they drink ABM coffee everyday
reaches 100 ( 20% of 500) or more in this survey, we can say without question that the
ABM Coffee Company is telling the truth.
But the number that the researchers got was 95, a proportion so close to the claim
that is 100. Do we reject the claim outright? Can we accept it? You know that it is possible
to get a different set of responses if we will get a different sample of 500 residents? In order
to make a correct decision, we need to set up the rejection region.
The REJECTION REGION is a range of values such that if the test statistic (Z, t, or p)
falls into that range, we decide to reject the H0 in favor of the H1. (Keller/Warrack,
p.324)
We will now illustrate how the rejection region is determined when =.05.
For a Left-Tailed Test (One-Tailed Directional Test), the rejection region will have an area
equal to 0.05 at the extreme left side of the Normal Curve. See the figure below.
The line that divides the Rejection Region and the Non-rejection Region (some books call it
Acceptance Region) corresponds to a Z value that we call Critical Value. This critical value
may be found in our Z-table. Since the rejection region (red region) has an area equal to
0.05, we have to look at our Z-table for the corresponding Z value. Were you able to locate
it? It’s exactly between -1.64 and -1.65. That means the Z value we are looking for is -
1.645.
REMEMBER: We use the Two-Tailed Test if we are using the symbol ≠ in the H1.
Do you understand? The preceding discussion is based only on =.05 level of significance.
In the beginning of this module, we talked about brands of coffee and the
people’s preference. When data are nominal, the only thing we can do to describe
the population or sample is to count the number of occurrences for each category.
From the counts we determine the proportions. (Keller/Warrack, p. 373)
̂ – p0
𝒑
𝑋
Z= where 𝑝̂ = 𝑛
√(p0q0/n) 𝑝̂ = sample proportion
EXAMPLE 1: A local government official claims that only up to 25% of all public
school students in the city own an electronic gadget that can be used for distance
learning like cellphone, tablet, or laptop. To test the claim, a group of Grade 11
Statistics students made a survey and found out that out of 1, 000 randomly
selected students, 275 indicated that they are ready for Online learning. Can we
infer from the data that the local official is true to his claim? Use =.05
SOLUTION:
(1) H0: The proportion of students who own an electronic gadget is at most 25%,
p0 ≤ 0.25
H1: The proportion of students who own an electronic gadget is more than 25%,
p0 > 0.25
NOTE: We are using the symbol > in our H1 because we hope to show that the
obtained sample 275 is significantly greater than 250, the 25% of 1,000.
(2) One-Tailed Test, =.05
(3) Is np0 > 5? YES! Is nq0 > 5? YES! np0 = (1000)(0.25)= 250 nq0
=(1000)(0.75)= 750
(4) USE Z TEST
𝑝̂ - p0 0.275 – 0.25
Z= = = 1.83 (computed
value)
(p0q0/n) [(0.25)(0.75)/1000)]
𝑋 275
𝑝̂ = = = 0.275 q0 = 1 – p0 = 1 – 0.25 = 0.75
𝑛 1000
1.83
̂ – p0
𝒑
𝑋
Z= where 𝑝̂ = 𝑛
√(p0q0/n) 𝑝̂ = sample proportion
x = number of successes
n = sample size
p0 = hypothesized population proportion
q0 = 1 – p0
Let us now solve the problem presented in the beginning of this module – the ABM coffee.
SOLUTION
(1) H0: The proportion of residents in Pinalagad that prefer the ABM coffee is 20%
or more, p0 ≥ 0.20
H1: The proportion of residents in Pinalagad that prefer the ABM coffee is
less than 20%, p0 < 0.20
NOTE: We are using the symbol < in our H1 because we hope to prove that the obtained
sample value 95 is significantly lesser than 100, the 20% 0f 500.
(2) One-Tailed Test, =.05
(3) Is np0 > 5? YES! Is nq0 > 5? YES! np0 = (500)(0.20)= 100 nq0 =(500)(0.80)=400
(4) USE Z TEST
𝑝̂ - p0 0.19 – 0.20
Z= = = -0.56 (computed value)
(p0q0/n) [(0.20)(0.80)/500)]
𝑋 95
𝑝̂ = = = 0.19 q0 = 1 – p0 = 1 – 0.20 = 0.80
𝑛 500
-0.56
(a) Set up the Rejection Region and the Critical Values for Left-Tailed, Right-Tailed,
and Two-Tailed tests. Use =.01.
(b) Set up the Rejection Region and the Critical Values for Left-Tailed, Right-Tailed,
and Two-Tailed tests. Use =.10.
Keller, Warrack. (2003). Statistics For Management and Economics. California USA:
Thomson Learning, Inc.
Levine, et. al. (2005). Statistics: A Handbook for Managers. New Jersey: Prentice Hall.
In this module we will tackle what are known as bivariate data. Bivariate data
involve two variables at a time. We will also learn the relationships between those
two variables. There are two types of bivariate data --- the categorical and the
numerical bivariate data. But in this module we will focus only on the numerical
bivariate data.
When drawing the scatter diagram we need the raw data from our two
variables. Each pair of observations from the two variables is represented by a dot.
This is similar to what you did in your Gen Math class when you plotted points on
the XY plane.
In most cases one variable seems to be dependent on the other variable. Just
to cite some examples --- an individual’s income somewhat depends on the number
of years of education (the higher your educational attainment, the higher you
expect your salary to become), a company’s sales depend on the amount spent in
advertising (This is the reason why companies spend a lot of money to advertise
their products), a student’s score in a major exam may depend on the number of
hours spent in studying (We sincerely hope you prepare really well for your exams).
EXAMPLE
It is unfortunate that this generation has experienced a dreaded
pandemic. The following are the actual number of Covid 19 cases in the Philippines
starting January 30, 2020 when the first case was detected. The data cover a
period of 20 weeks from January 30 to June 30, 2020 (en.m.wikipedia.org)
X 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Y 1 2 3 3 5 140 380 2084 4195 5878 6981 8488 10610 12305 13777
16 17 18 19 20
18086 21340 24787 27799 37514
Scatter Diagram
40000
Number of Covid 19
20000
Cases
0
0 5 10 15 20 25
Week 1 to Week 20
What do you notice about the general direction of the dots in our example?
Take a look at the Example again. You can clearly see that as time passes by,
the number of Covid 19 cases also increases. As the independent variable increases,
the dependent variable also increases. When this happens we say that there is a
positive linear relationship between the two variables.
In a certain senior high school in Valenzuela City where Gen Math (X) and Statistics (Y) are
offered as core subjects, a sample of 15 students was drawn. The midterm grades for both
subjects were recorded for each student. The data are listed below. (Keller/Warrack, p. 67)
X 65 60 93 68 74 81 60 85 88 75 63 79 80 60 72
Y 74 72 84 71 68 85 63 73 79 65 62 71 74 68 73
100
STATISTICS 50
0
0 20 40 60 80 100
GEN MATH
SOLUTION: (a)
(b) There is a positive linear relationship between the two core subjects, and
the relationship is of medium-strength.
X 2 2 2 3 3 4 5 5 6 6
Y 57 63 70 72 69 75 73 84 82 89
b. Describe the relationship between the two variables with respect to direction and
1. Choose the scatterplot that best fits this description: "There is a strong,
positive, linear association between the two variables." Explain each choices
why or why not it is a solution to the problem. (Khan Academy)
Belecina, R., Baccay, E., & Mateo, E. (2016). Statistics and Probability. Manila,Philippines:
REX Book Store Inc.
Keller, Warrack. (2003). Statistics For Management and Economics. California USA:
Thomson Learning, Inc.
Levine, et. al. (2005). Statistics: A Handbook for Managers. New Jersey: Prentice Hall.
Do this Pre-Test: Write True if the statement is correct, False otherwise. Write
your answer in your notebook.
____________1.) 1.001 can be a representation of correlation coefficient r ?
____________2.) Negative relationship means direct relationship.
____________3.) The first step in computing Pearson’s sample correlation coefficient r
is to get the sum of all entries in all columns.
____________4.) If the coefficient of correlation falls between 0.51 to 0.74, there is a
high negative correlation.
____________5.) In the Pearson r, n represents sum of x-values.
In this lesson, the learners will learn how to calculate the Pearson’s Sample
Correlation Coefficient. They will also learn how solve problems involving
correlation analysis diagram.
The Correlation coefficient, r, between sets of the data is a measure of how well
they are related. It is a measure of the strength of the relationship between or among
variables.
Solution:
‘Since the table is already completed, proceed to substitution for the values required
for the formula:
Below are the data for six participants giving their number of years in college
(X) and their subsequent monthly income (Y). Which one of the following best
describes the correlation between X and Y?
Income (y) 15 15 20 25 30 35
Solution:
Step 1: Complete the table
0 15 0 225 0
1 15 1 225 15
3 20 9 400 60
4 25 16 625 100
4 30 16 900 120
6 35 36 1225 210
∑ 𝒙 = 𝟏𝟖 ∑ 𝒚 = 𝟏𝟒𝟎 ∑ 𝒙𝟐 = 𝟏𝟒𝟎 ∑ 𝒚𝟐 = 3600 ∑ 𝒙𝒚 = 𝟓𝟎𝟓
Step 3: Interpret
Activity 1. Complete the table below. Fill in the blanks in the formula to arrive at the
computed Pearson r. Then interpret the result.
X Y XY X2 Y2
15 5 225
23 3
11 8 64
9 10 100
15 8 64
20 20 400
∑X = ∑Y = ∑ 𝒙𝟐 = ∑ 𝒚𝟐 = ∑ 𝒙𝒚 =
x y x2 y2 xy
5 25 25 635 125
3 20 9 400 60
4 21 16 441 84
10 35 100 1225 350
15 38 225 1444 570
∑ X =37 ∑ Y =139 ∑ 𝒙𝟐 = (1)____ ∑ 𝒚𝟐 = (2)____ ∑ 𝒙𝒚 = (3)____
Belecina, R., Baccay, E., & Mateo, E. (2016). Statistics and Probability. Manila,Philippines:
REX Book Store Inc.
Bluman, A. (2018). Elementary Statistics: A Step by Step Approach 10th edition. McGraw
Hill. New York, USA.
Canlapan, R. (2016). Statistics and Probability. Makati, Philippines: Diwa Learning System
Inc.
Keller, Warrack. (2003). Statistics For Management and Economics. California USA:
Thomson Learning, Inc.
Levine, et. al. (2005). Statistics: A Handbook for Managers. New Jersey: Prentice Hall.
PERCDC Learnhub
Walpole, R., Myers, R., Myers, S., and Ye, K., (2012). Probability and Statistics for Engineers
and Scientists 9th edition. Pearson Education Inc. Massachusetts, USA.
Do this Pre-Test: Write True if the statement is correct, False otherwise. Write
your answer in your notebook.
____________1.) The y-intercept is the value of y when x=0.
____________2.) correlation is used to determine the existence, strength, and direction
of relationship between bivariate data?
____________3.) In regression analysis, a response variable is also known as the
dependent variable.
____________4.) The equation for the straight line that is used to estimate y based on
x is referred to as linear equation.
____________5.) Independent Variable is also known as output variable.
Let’s have some review of the Dependent and Independent Variables. By example
below, you may be reminded of what is meant by Dependent Variable as the values
that predicts or assumes the predictor and sometimes called the outcome or response
variable:
The technique used to develop the equation for a straight line and make predictions
about relationship of two variable is called Regression Analysis. The equation for
the straight line that is used to estimate y based on x is referred to as regression
equation.
The equation of the regression line is written as: 𝑦 = 𝑎 + 𝑏𝑥, 𝑎 = 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡, 𝑏 =
𝑠𝑙𝑜𝑝𝑒.
The formulas used to generate the Regression Equation (least square method) are:
𝑛(∑ 𝑥𝑦)−(∑ 𝑥)(∑ 𝑦) ∑𝑦 ∑𝑥
𝑏= 2 2 𝑎= − 𝑏( )
𝑛(∑ 𝑥 )−(∑ 𝑥) 𝑛 𝑛
Example 1:
The point estimate of graduate GPA when EER is 85 is: 𝑦 = 3.181 − 0.014(85) =
1.99
Regression analysis is a powerful statistical method that allows you to examine the
relationship between two or more variables of interest. While there are many types
of regression analysis, at their core they all examine the influence of one or more
independent variables on a dependent variable.
The time x in years that an employee spent at a company and the employee’s hourly
pay, y, for 5 employees are listed in the table below:
x y x2 y2 xy
5 25 25 635 125
3 20 9 400 60
4 21 16 441 84
10 35 100 1225 350
15 38 225 1444 570
37 139 375 4135 1189
In the problem in the Assessment part, how much could be employee’s hourly pay if
he is already 20 years in the company? Round off your answer to the nearest whole
number.
Belecina, R., Baccay, E., & Mateo, E. (2016). Statistics and Probability. Manila,Philippines:
REX Book Store Inc.
Bluman, A. (2018). Elementary Statistics: A Step by Step Approach 10th edition. McGraw
Hill. New York, USA.
Canlapan, R. (2016). Statistics and Probability. Makati, Philippines: Diwa Learning System
Inc.
Keller, Warrack. (2003). Statistics For Management and Economics. California USA:
Thomson Learning, Inc.
Levine, et. al. (2005). Statistics: A Handbook for Managers. New Jersey: Prentice Hall.
PERCDC Learnhub
Walpole, R., Myers, R., Myers, S., and Ye, K., (2012). Probability and Statistics for Engineers
and Scientists 9th edition. Pearson Education Inc. Massachusetts, USA.