Test on Population Proportion

Learner's Module in Statistics and

Probability
Quarter 4 ● Module 5 ● Weeks 5 – 6

http://www.sixsigmaterial.com

ANNE D. ORTIGA
Developer
Department of Education • Cordillera Administrative Region
 What I Need to Know

Hello learner! This module was prepared and written with

you in mind. Its scope is to equip you with essential knowledge
and skills in order for you to understand and explain concepts on
random sampling and related topics and then apply them in
solving real life situations.

While going through this module, you are expected to:

 formulates the appropriate null and alternative hypotheses on
a population proportion;
 identifies the appropriate form of the test-statistic when the
Central Limit Theorem is to be used;
 identifies the appropriate rejection region for a given level of
significance when the Central Limit Theorem is to be used;
 compute for the test statistic value (population proportion);
 draw conclusion about the population proportion based on
the test-statistic value and the rejection region; and
 solve problems involving test of hypothesis on the population
proportion.

What I Know

This pre-test will determine your prior knowledge of the topic. 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
before the number.

is the formula for z – test of proportion when you are

1. What
comparing the sample proportion with population proportion?
2. If and what is ?
A. 0.29 B. 0.30 C. 3.42 D. 3.43
3. If what is ?
A. 0.43 B. 0.47 C. 0.57 D. 0.67
4. In a one sample z – test of proportions, the computed z – value
lies in the rejection area. What does this mean?
A. The sample proportion is equal to the hypothesized proportion.
B. The sample proportion is equal to the population proportion.
C. The sample proportion is not equal to the hypothesized
proportion.
D. The sample proportion is not equal to the population
proportion.
5. In a one – tailed z – test of proportions, the comparative
statement between the computed value and critical value is 0.35
< 0.42. What decision should be made about ?
A. Reject C. Neither accept nor reject
B. Retain D. None of the above.
6. When the null hypothesis is retained which of the following is
true? A. There is sufficient evidence to back up the decision.
B. There is no sufficient evidence to back up the decision.
C. The conclusion is guaranteed.
D. The conclusion is not guaranteed.
For items 7 – 11, refer to the problem below:
A researcher claimed that more than 55% of Grade 11 students in
the city have internet at home. In a sample of 150 grade 11
students, 90 are found to have internet at home. Use

7. What is the null and alternative hypothesis in the problem

above?
A. C.
B. D.
8. What is the level of significance?

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 A.0.05 B. 0.1 C. 0.01 D. 0.025
9. Compute the test statistic value.
A. 0.45 B. 0.55 C. 1.23 D. 1.32
10. Determine the critical value.
A. – 1.645 B. 1.645 C. ±1.645 D. ±1.96

11. 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.

For items 12 – 15, refer to the problem below:

Mr. Bautista asserts that fewer than 5% of the bulbs that he sells
are defective. Suppose 100 bulbs are randomly selected, each are
tested and 2 defective bulbs are found. Does this provide sufficient
evidence for Mr. Bautista that the fraction of defective bulbs is
less than 0.05? Use

12.What is the null and alternative hypothesis in the problem?

A. C.
B. D.
13. Compute the test statistic value.
A. – 1.37 B. – 1.38 C. – 2.14 D. – 2.15
14. Determine the critical value.
A. – 1.28 B. 1.28 C. – 1.645 D. 1.645

15. Based on the computed value and critical value, is Mr.

Bautista correct in his assertion that fewer than 5% of the
bulbs that he sells are defective?
A. Yes B. No C. Undecided D. Cannot be determined

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 What’s In

Let us see how much you can recall the steps in conducting
hypothesis testing by arranging the following steps
chronologically. Write 1 – 6 on the space before each item being 1
as the first step and 6 as the last step.
_____ State the null and alternative hypotheses.
_____ Draw a conclusion.
_____ Determine the critical value.
_____ Determine the level of significance.
_____ Select the appropriate test statistics.
_____ Compute the test – statistic value.

What’s New

There are some certain situations when the data that is being
analyzed involves proportions or percentages. Real life examples
of such data are the following: (1) a store owner wants to know if
his customers will patronize a new product, (2) a teacher wants to
know the percentage of her students who joins in their online
class, and (3) a grade 12 student wants to know what percentage
of senior high school students in his school chose online as
learning modality.

If you were asked by the people mentioned above to help them,

what test statistic would you recommend in computing for the
test- statistic value? What are the steps would you tell them to do?
To answer these questions, read and understand the lessons in
this module.

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 What Is It

The steps in testing the hypothesis involving population

proportion is the same as the steps in testing the hypothesis
involving population means. The difference is that, in testing
population proportion, we use z – test as test statistic no matter
what our sample size is unlike in testing the means where you
either use z – test or t – test depending on the given sample size
and standard deviation.

The - test of proportion is the tool used in hypothesis testing

concerning proportions or percentages. In testing hypothesis
concerning proportions or percentages, we must make the
following assumptions:

1. The conditions for a binomial experiment are met which is,

there is a fixed number of independent trials with constant
probabilities, and each trial has two outcomes that is
usually classify as “success” and “failure.”
2. The condition and are both satisfied.

The one – proportion z – test is used when you are testing

one sample proportion. The formula is:

where:

The two – proportion z – test is used when you are testing two
sample proportions. The formula is:

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where:

Sometimes, the sample proportion, is directly given such

as “55% of the respondents” which may be written as
When it is not directly given, we need to calculate it using the
formula:

where:

For example, the statement “30 out of 50 respondents were

interviewed.” The computed value of the sample proportion is

For you to be able to test hypothesis involving population

proportion, study the following examples:

Example 1: Test the claim that 22% of senior medical students of

a certain medical school prefer pediatrics. Sample data
consist of 100 randomly selected seniors of the medical
school, with 20 of them choosing pediatrics. Use 5% level
of significance.

Solution:
Step 1: State the null and alternative hypotheses.
(two-tailed test)
Step 2: Determine the level of significance.

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Step 3: Select the test statistic. Since we are testing one sample

proportion, we use .
Step 4: Computation of the test statistic value. Since the sample
proportion is not directly given, we must solve it first, so
with .

and

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 0.5.

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 Use the Areas Under the Normal Curve Table. The area
0.4750 is under column headed 6. Move along this row to the
left until 1.9 under column headed is reached. Therefore,
At 5% level of significance, the critical value is .
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 Non -rejection

Rejection Region Region

Rejection Region

� �

-1.96 1.96
Critical Value Critical Value

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

, falls within the non – rejection region, retain the
null hypothesis. Conclude that there is sufficient sample
evidence to support the claim that 22% of senior medical
students prefer pediatrics. This result is significant at
level.

Example 2: Based on registered birth last 2018, PSA said that

the rate of males being born is 52.2%. Suppose a
Statistics student believed that the rate of male babies
born increased in 2019. He took a random sample from
the different barangays in their municipality and
determined the rate of males born of the same year. The
results revealed that there were 452 out of 825 male
babies born last year. Can it be concluded that the rate
of males born has increased? Use 0.10 level of
significance.

Solution:
Step 1: State the null and alternative hypotheses.
(one-tailed test)
Step 2: Determine the level of significance.
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Step 3: Select the test statistic. Since we are testing one sample

proportion, we use .

Step 4: Computation of the test statistic value. Since the sample

proportion is not directly given, we must solve it first, so
with .
548 and 78

Step 5: Determine the critical value. The alternative hypothesis is

directional. Hence, the one-tailed test shall be used. This
time, will not be divided by 2. Subtract 0.10 from 0.5.

Use the Areas Under the Normal Curve Table. The area 0.40
is between 0.3997 under column headed 8 and 0.4015 under
column headed 9. Move along this row to the left until 1.2 under
column headed is reached. Therefore, At 10% level of
significance, the critical value is .

Non-rejection Rejection Region

region
�

1.28
Critical Value
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Step 6: Draw a conclusion. Since the computed test statistic
value, , falls within the rejection region or
, we reject the null hypothesis. Therefore,
there is enough evidence to support the claim that the rate
of males being born in 2019 increased.

Example 3: A sample survey of a radio program in Baguio City

shows that 110 out of 250 men and 80 out of 200
women dislike the same program. A researcher wants to
know whether the difference between the two sample
proportions is significant or not at 0.10 level of
significance.

Solution:
Step 1: State the null and alternative hypothesis.
(two-tailed test)
Step 2: Determine the level of significance.

Step 3: Select the test statistic. Since two sample proportion are
being compared, we use value for two sample proportion:

Step 4: Computation of the test statistic value. Since the

proportions of the two sample groups were not given
directly, we solve it first including and .

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Step 5: Determine the critical value: The alternative hypothesis is
non-directional. Hence, the two-tailed test shall be used.
Divide by 2, and then subtract the quotient from 0.50.

Use the Areas Under the Normal Curve Table. The area 0.45
is between 0.4495 under column headed 4 and 0.4505 under
column headed 5. Move along this row to the left until 1.6 under
column headed is reached. Therefore, At
10% level of significance, the critical value is
.

Rejection
Rejection
Non -rejection
Region
Region Region
� �

-1.645 1.645

Critical Value Critical Value

Step 6: Draw a conclusion: Since the computed test statistic

value is not greater than , do not reject the null
hypothesis. Conclude that there is no significant difference
between the proportion of male and proportion of female in
disliking the same program.

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Example 4: In a study of 22,000 male adults, half were given
regular doses of aspirin while the other half were given
placebos with no effects. Among those who took placebos,
189 suffered heart attacks. Among those who took
aspirins, 104 suffered heart attacks. At 0.01 level of
significance, test the claim that the aspirin group has
significantly lower heart attack rate than those placebo
group.

Solution:
Step 1: State the null and alternative hypothesis.
(one-tailed test)
Step 2: Determine the level of significance.
Step 3: Select the test statistic. Since two sample proportion are
being compared, we use value for two sample proportion:

.
Step 4: Computation of the test statistic value. Since the
proportions of the two sample groups were not given
directly, we solve it first including and .

Step 5: Determine the critical value. The alternative hypothesis is

directional. Hence, the one-tailed test shall be used.
Subtract from 0.50.
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 Use the Areas Under the Normal Curve Table. The z – value that
corresponds to the area which is nearest to the area 0.49 is 2.3 3.

Rejection
N on -rejection
Region
R egion
�

-2.33
Critical Value

„
Step 6: Draw a conclusion. Since the computed test statistic value
is greater than the absolute value of , reject the null
hypothesis. Conclude that there is sufficient evidence to
indicate that the aspirin group has significantly lower heart
attack rate than those placebo group.

What’s More

Activity 1. Compute the test statistic value

Direction: Choose the appropriate test statistic to be used based

on the given information then compute. Use a sheet of paper for
your solution.

1. Given:
2. Given:

3. Given:

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 4. Given:

Activity 2. Hypothesis testing on proportions!

Direction: Complete the table below. The first one is done for you.
1-
tailed/
Decision
2-
tailed
1. 0.01 1-tailed Reject
2.
3. 0.05 1-tailed
4.
5. 0.10

Activity 3: Problem Solving

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

1. A garment factory distributes two brands of jeans. If it is found

that 75 out of 250 customers prefer brand A and that 30 out of
150 prefer brand B, can we conclude at 0.05 level of
significance that brand A outsells brand B?

2. A SHS student conducted a survey to test the claim that “less

than half of all the adults are annoyed by the violence on
television”. Suppose that from a poll of 2,400 surveyed adults,
1,152 indicated their annoyance with television violence. Test
this claim using 0.10 level of significance.

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 What I Have Learned

To check your understanding of the lesson, complete the

following statements:

1. In a one – tailed test of proportion, the null hypothesis is

rejected when _________________________________________________.
2. The is used when__________________________.
The is used when___________________________.

What I Can Do

Let’s apply!
The survey done by Pulse Asia (February 22 – March 3, 2021)
revealed that the top two brands of COVID-19 vaccine prefer by
Filipinos are Pfizer and Sinovac. You are part of a team to make a
survey on which brand of COVID-19 vaccine the residents in
your barangay prefer. It was revealed that ____ (choose a
number which is more than half of the number of residents in
your barangay) out of ____ (approximate the number of resident
in your barangay) residents in the barangay prefer the Pfizer
brand. Conduct a hypothesis testing on the proportion of the
preferred COVID 19 vaccine brand. Use

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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
before the number.

1. Whatis the formula for z – test of proportion when you are

comparing the two sample proportions?

2. If and what is ?
A. 0.29 B. 0.30 C. 0.42 D. 0.43
3. If what is ?
A. 0.43 B. 0.47 C. 0.57 D. 0.67
4. In a one sample z – test of proportions, the computed z – value
lies in the non – rejection region. What does this mean?
A. The sample proportion is equal to the hypothesized
proportion.
B. The sample proportion is equal to the population proportion.
C. The sample proportion is not equal to the hypothesized
proportion.
D. The sample proportion is not equal to the population
proportion.
5. In a one – tailed z – test of proportions, the comparative
statement between the computed value and the critical value is
0.35 = 0.35. What decision should be made about ?
A. Reject C. Neither accept nor reject
B. Retain D. None of the above.
6. When the null hypothesis is rejected, which of the following is
true? A. There is sufficient evidence to back up the decision.
B. There is no sufficient evidence to back up the decision.
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 C. The conclusion is guaranteed.
D. The conclusion is not guaranteed.

For items 7 – 11, refer to the problem below:

A researcher claimed that more than 55% of Grade 11 students
in the city have internet at home. In a sample of 150 grade 11
students were selected at random and 85 are found to have
internet at home. Use

7. What is the null and alternative hypothesis in the problem

above?
A. C.
B. D.

8. What is the level of significance?

A. 0.05 B. 0.1 C. 0.01 D. 0.025
9. Compute the test statistic value.
A. 0.45 B. 0.55 C. 1.23 D. 1.32
10. Determine the critical value.
A. – 1.645 B. 1.645 C. – 1.28 D. 1.28

11. 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.

For items 12 – 15, refer to the problem below:

Mr. Bautista asserts that fewer than 5% of the bulbs that he
sells are defective. Suppose 100 bulbs are randomly selected,
each are tested, and 2 defective bulbs are found. Does this
provide sufficient evidence for Mr. Bautista that the fraction of
defective bulbs is less than 0.05. Use

12. What is the null and alternative hypothesis in the problem?

19
A. C.
B. D.

13. Compute the test statistic value.

A. – 2.15 B. – 2.14 C. – 1.38 D. – 1.37
14. Determine the critical value.
A. – 1.96 B. 1.96 C. – 1.645 D. 1.645

15. Base on the computed test statistic value and the critical
value, is Mr. Bautista correct from his assertion that fewer
than 5% of the bulbs that he sells are defective?
A. Yes B. No C. Undecided D. Cannot be determined

Additional Activity

These additional problems are prepared for you to have more

practice on conducting hypothesis testing involving population
proportion.

Direction: In the following problem, (a) state the null and

alternative hypothesis, (b) select and compute the test statistic, (c)
determine the critical value and the rejection region, and (d) draw
a conclusion.

1. It is claimed that 70% of SHS students in a certain district

have internet at home. A survey among 1500 SHS students
in that district revealed that 1025 have internet at home.
Use 0.05 level of significance

2. Mr. Santos, a garment store owner, claimed that the brand

of sweatpants his customers prefer is Brand A compared to
Brand B. He conducted a survey of 100 customers. It was
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 revealed that 55 prefer Brand A and the rest prefer Brand B.
At 5% level of significance test the claim of Mr. Santos that
the brand of sweatpants his customers prefer is Brand A
compared to Brand B.
ANSWER KEY

REFERENCES

Books:

Belecina, Rene et al. 2016. Statistics and Probability, 268-281.

Rex Book Store, Inc.

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Pagoso, Cristobal et al. 1992. Fundamental Statistics for College
Students, 201-209. Sinag – Tala Publishers

Online sources:

https://www.youtube.com/watch?v=qdM16bfNSyE
Accessed: May 13, 2021

https://psa.gov.ph/vital-statistics/id/163858
Accessed: May 13, 2021

https://www.youtube.com/watch?v=pCbNUnZ98oE
Accessed May 13, 2021

https://www.youtube.com/watch?v=76VruarGn2Q

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