Statistics-and-Probability G11 Quarter-4 Module-6 Computation-of-Test-Statistic-on-Population-Mean final PDF

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Statistics andProbabilityQuarter 4 – Module 6:Computing Test Statistic onPopulation MeanDevelopment Team of the Module Writer: Nelda L. Oabel Editors: Jerome A. Chavez, Gilberto M. Delfina, Garry S. Villaverde, and Pelagia L. Manalang Reviewers: Josephine V. Cabulong, and Nenita N. De Leon Illustrat...

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Statistics and Probability Quarter 4 – Module 6: Computing Test Statistic on Population Mean

Statistics and Probability – Grade 11 Alternative Delivery Mode Quarter 4 – Module 6: Computing Test Statistic on Population Mean First Edition, 2020 Republic Act 8293, Section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them. Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio Development Team of the Module Writer:

Nelda L. Oabel

Editors: Jerome A. Chavez, Gilberto M. Delfina, Garry S. Villaverde, and Pelagia L. Manalang Reviewers: Josephine V. Cabulong, and Nenita N. De Leon Illustrator: Jeewel C. Cabriga Layout Artist: Edna E. Eclavea Management Team: Wilfredo E. Cabral, Regional Director Job S. Zape Jr., CLMD Chief Elaine T. Balaogan, Regional ADM Coordinator Fe M. Ong-ongowan, Regional Librarian Aniano M. Ogayon, Schools Division Superintendent Maylani L. Galicia, Assistant Schools Division Superintendent Randy D. Punzalan, Assistant Schools Division Superintendent Imelda C. Raymundo, CID Chief Generosa F. Zubieta, EPS In-charge of LRMS Pelagia L. Manalang, EPS

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Statistics and Probability Module 6: Computing of Test Statistic on Population Mean

Introductory Message For the facilitator: Welcome to the Statistics and Probability for Senior High School Alternative Delivery Mode (ADM) Module on Computing Test Statistic on Population Mean! This module was collaboratively designed, developed, and reviewed by educators both from public and private institutions to assist you, the teacher or the 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:

Notes to the Teacher This contains helpful tips or strategies that will help you in guiding the learners. 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 Senior High School Alternative Delivery Mode (ADM) Module on Computing Test Statistic on Population Mean! The hand is one of the most symbolical parts 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 as a learner, you are 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. This module has the following parts and corresponding icons: What I Need to Know

What I Know

What’s In

What’s New

This will give you an idea of the skills or competencies you are expected to learn in the module. This part includes an activity that aims to check what you already know about the lesson to take. If you get all the answers correct (100%), you may decide to skip this module. This is a brief drill or review to help you link the current lesson with the previous one. In this portion, the new lesson will be introduced to you in various ways such as a story, a song, a poem, a problem opener, an activity, or a situation.

What Is It

This section provides a brief discussion of the lesson. This aims to help you discover and understand new concepts and skills.

What’s More

This comprises activities for independent practice to solidify your understanding

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and skills of the topic. You may check the answers to the exercises using the Answer Key at the end of the module. What I Have Learned

This includes questions or blank sentences/paragraphs to be filled in to process what you learned from the lesson.

What I Can Do

This section provides an activity which will help you transfer your new knowledge or skill into real life situations or concerns.

Assessment

This is a task which aims to evaluate your level of mastery in achieving the learning competency.

Additional Activities

In this portion, another activity will be given to you to enrich your knowledge or skill of the lesson learned. This also aims for retention of learned concepts.

Answer Key

This contains answers to all activities in the module.

At the end of this module, you will also find:

References

This is a list of all sources used in developing this module. The following are some reminders in using this 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! iv

What I Need to Know In this module, you will learn how to compute the test statistic on a population mean particularly the t-test and z-test. It is a skill that you need to develop to be able to determine whether you reject the null hypothesis or otherwise (to be discussed in the next module). Perform each activity independently. If you find any difficulty in answering the exercises, you may ask the assistance of your teacher or you may consult your peers. After going through this module, you are expected to: 1. determine the appropriate test statistic to be used in the given problem/situation; and 2. compute for the test statistic value (population mean).

What I Know Directions: Choose the best answer to the given questions or statements. Write the letter of your choice on a separate sheet of paper. 1. It refers to a value calculated from sample data which is needed in

deciding whether the null hypothesis is rejected or not. A. test statistics C. null hypothesis B. critical region D. alternative hypothesis 2. What test statistic will be used if the sample size is above 30?

A. t-test B. z-test

C. population mean D. standard deviation

3. What test statistic can be used when the population standard deviation

is known? A. t-test B. z-test

C. population mean D. standard deviation

4. What test statistic can be used when the population standard deviation

is unknown? A. t – test B. z – test

C. population mean D. standard deviation 1

5. When finding the z-computed value, which formula should be used for

hypothesis testing? A. 𝑧 =

𝑥−𝜇  𝜎 √𝑛

B. 𝑧 =

𝑥−𝜇 

C. 𝑧 =

𝑠 √𝑛

𝜇− 𝑥 𝑠 √𝑛

D. 𝑧 =

𝜇− 𝑥 𝜎 √𝑛

6. When should you use the t-test?

I. When you are testing for a population mean II. When the sample standard deviation is given III. When the population standard deviation is given IV. When you are testing a proportion/percentage of a population A. I and II

B. II and III

C. I and III

D. II and IV

For nos. 7-8, refer to the problem below: Given: 𝐻𝑜 : μ = 8.6

𝐻𝑎 : μ > 8.6

The study has a sample mean of 9.1 and a standard deviation of 2.1 conducted among 25 respondents. Use 𝛼 = 0. 05. 7. What test statistics should be used?

A. t-test B. z-test

C. population mean D. standard deviation

8. What is the computed value?

A. -1. 190

B. – 0. 567

C. 0. 567

D. 1.190

9. How many samples are best when dealing with z-test?

A. cannot be determined B. exactly 30

C. smaller than 30 D. equal or larger than 30

For nos. 10 – 12, refer to the given problem below: The Choco Toppings, Inc. is one of the manufacturers of chocolate toppings which uses a machine to dispense liquid ingredients into bottles that move along a filling line. The owner claims that the machine can dispense at an average of 50 grams with a standard deviation of 0.7 grams. A sample of 35 bottles was selected and it was found out that the average amount dispensed in the sample is 49.3 grams. Test the claims of the owner of the company at 5% level of significance. 10. Which of the following information is correct?

A. 𝛼 = 0.5

B. 𝜎 = 0.7

C. 𝑥 = 35

D. 𝜇 = 49.3

11. What test statistic will be used?

A. t-test B. z-test

C. population mean D. standard deviation 2

12. Find the computed value.

A. -5.916

B. -4.950

C. 4.950

D. 5.916

13. Which test statistic will be used if the sample size is 15?

A. t-test B. z-test

C. cannot be determined D. neither t-test nor z-test

14. Which statistical method can you use when you have a normal

distribution of data? A. t–test only B. z–test only

C. either t–test or z–test D. neither t–test nor z–test

15. A tire manufacturer claims that its tires have a mean life of 40,000 km. A

random sample of 46 of these tires is tested and the sample mean is 38,000 km. Assume that the population ’s standard deviation is 2,000 km and the lives of the tires are approximately normally distributed. Determine the computed value at 5% level of significance. A. -6.782 B. -3.033 C. 3.033 D. 6.782

How do you find this pre-test? Did you encounter both familiar and unfamiliar terms? Kindly compare your answer in the Answer Key on the last part of this module If you obtain 100% or a perfect score, skip the module and immediately move to the next module. But if you missed a point, please proceed with the module as it will enrich your knowledge in computing the test statistic.

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Lesson

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Computing Test Statistic on Population Mean

One of the steps in hypothesis testing is the computation of test statistic. Remember that it is the value calculated from a sample data which is needed whether you reject the null hypothesis or not. Do you still remember when to use t-test? How about z-test? Answer the activity that follows for a short review on t-test and z-test.

What’s In

Is It T or Z? Directions: Identify the appropriate test statistic to be used based on the given information. Write T if it is t-test and Z if it is z-test. 1. The sample mean is 345 and the sample size is 46. The population is normally distributed with a standard deviation of 11. Test the hypothesis at 0.05 level of significance. Consider the hypotheses below: 𝐻𝑜 : 𝜇 = 342 𝐻𝑎 : 𝜇 ≠ 342 2. Test at 𝛼 = 0.05 the null hypothesis 𝐻𝑜 : 𝜇 = 2. 19 against the alternative hypothesis 𝐻𝑎 : 𝜇 < 2. 19 with 𝑛 = 18, 𝑥 = 1.36, and 𝑠 = 0.14. Assume that the population is approximately normal. 3. The sample size is less than 18 and the standard deviation is 3. 67. 4. 𝑥 = 125.3

𝑠=5

𝜇 = 124

5. 𝑥 = 25.4

𝜇 = 22.6

𝜎 = 15

𝑛 = 24 𝑛 = 118

𝛼 = 0.05 𝛼 = 0.01

Were you able to answer all the questions correctly? If yes, the next activity will be easy for you. If not, go back your notes about the test concerning means.

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What’s New t-Test vs z-Test Directions: Complete the diagram below. Do you know the standard deviation (σ)?

YES

NO

1.Use:

Is the sample size above 30?

__________

YES

NO

2. Use:

3. Use:

__________

__________

I think you are very much ready for this topic. Read, analyze, and study the given examples carefully.

What Is It There are two specific test statistics used for hypothesis testing concerning means: z-test and t-test. If the sample size is large, where 𝑛 ≥ 30 and the population standard deviation (𝜎) is known, use z-test. In finding the z-value, use the formula below: 𝑥 − 𝜇 𝑧= 𝜎 √𝑛 where:

𝑥 = sample mean 𝜇 = population mean 𝑛 = sample size 𝜎 = population standard deviation 5

On the other hand, t- test is used when 𝑛 < 30, the population is normal or nearly normal, and sample standard deviation (𝑠) is unknown. The formula for the t- value is: 𝑡= where:

𝑥 − 𝜇 𝑠 √𝑛

𝑥 = sample mean 𝜇 = population mean 𝑛 = sample 𝑠 = sample standard deviation

The degrees of freedom is 𝑛 − 1 or 𝑑𝑓 = 𝑛 − 1. Study the following examples. Example 1: Compute the z-value given the following information. Use onetailed test and 0. 05 level of significance. 𝑥 = 70 𝜇 = 71.5 𝜎=8 𝑛 = 100 Solution: Since σ is known and n ≥ 30, we will use z-test. Thus, we have: 𝑧=

𝑧=

𝑧= 𝑧=

𝑥 − 𝜇 𝜎 √𝑛 71. 5 − 70 8 √100

Use the formula for z-test.

Substitute the given value to the formula.

1.5 8 10

Simplify.

1.5 0.8

𝐳 = 𝟏. 𝟖𝟕𝟓

Therefore, the computed z-value is 1.875.

Example 2: In the first semester of the school year, a random sample of 200 students got a mean score of 81.72 with a population standard deviation of 15 in Statistics and Probability test. The population mean is 79.83. Use 0.05 level of significance. Solution: To answer the problem, let us first identify the given. We have: 𝑥 = 81.72 𝜇 = 79.83 𝜎 = 15 𝑛 = 200 Since σ is known and n ≥ 30, we will use z-test.

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𝑧=

𝑧=

𝑥 − 𝜇 𝜎 √𝑛 81.72 − 79. 83 15 √200

𝑧=

1. 89 15 14. 14

𝑧=

1. 89 1.06

𝐳 = 𝟏. 𝟕𝟖𝟑

Use the formula for z-test.

Substitute the given value to the formula. Simplify.

Therefore, the computed z-value is 1.783.

In Central Limit Theorem, the sample standard deviation (𝑠) may be used as an estimate of the population standard deviation (𝜎) when the value of 𝜎 is unknown. Consider the given examples below: Example 3: In the past, the average length of an outgoing call from a business office has been 140 seconds. A manager wishes to check whether that average has decrease after the introduction of policy changes. A sample of 150 telephone calls produced a mean of 135 second, with a standard deviation of 30 seconds. Perform the relevant test at 1% level of significance. Solution: Let us first identify the given. We have: 𝑥 = 135 𝜇 = 140 𝑠 = 30 𝑛 = 150 Since n ≥ 30, we will use z-test by replacing 𝝈 with its estimate s. 𝑧=

𝑥 − 𝜇 𝜎 √𝑛

Use the formula for z-test.

135 − 140 30 √150

Substitute the given value to the formula.

𝑧=

−5 30 12.25

Simplify.

𝑧=

−5 2.45

𝑧=

𝐳 = − 𝟐. 𝟎𝟒𝟏

Therefore, the computed z – value is -2.041.

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Example 4: Compute the t-value given the following information: 𝑥 = 129.5 𝜇 = 127 𝑠=5 𝑛 = 12 Solution: Since σ is unknown and n < 30, we will use t-test. Thus, we have: 𝑡=

𝑥 − 𝜇 𝑠 √𝑛

Use the formula for t-test.

129. 5 − 127 5 √12

Substitute the given value to the formula.

𝑡=

2. 5 5 3.46

Simplify.

𝑡=

2.5 1.44

𝑡=

Therefore, the computed t – value is 1. 736.

𝐭 = 𝟏. 𝟕𝟑𝟔

Example 5: The government claims that the monthly expenses of a Filipino family with four members is P10,000. A sample of 26 family’s expenses has a mean of P10,900 and a standard deviation of P1,250. Is there enough evidence to reject the government’s claim at 𝛼 = 0. 01? Solution: Let us first identify the given, so we have: 𝑥 = P10,900 𝑡=

𝑡=

𝑥 − 𝜇 𝑠 √𝑛

10 900 − 10 000 1 250 √26

𝑡=

900 1 250 5.10

𝑡=

900 245. 10

𝐭 = 𝟑. 𝟔𝟕𝟏

𝜇 = P10,000

𝑠 = P1,250

𝑛 = 26

Use the formula for t-test.

Substitute the given value to the formula. Simplify.

Therefore, the computed t-value is 3.671.

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Now, it’s your turn to answer the following exercises.

What’s More Activity 1: Find My z-Value! Directions: Find the computed z-value of the following. Write your answer to the nearest thousandths. Show your solutions.

1. 𝑥 = 21. 75

𝜇 = 20. 83 𝜎 = 2.75 𝑛 = 38

4. 𝑥 = 45 000

𝜇 = 46 100 𝜎 = 1 795 𝑛 = 50

2. 𝑥 = 11. 23

𝜇 = 12. 01 𝜎 = 3.0 𝑛 = 44

5. 𝑥 = 1.72

𝜇 = 1.83 𝜎 = 1.05 𝑛 = 36

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3. 𝑥 = 891.75

𝜇 = 890. 25 𝜎 = 11.75 𝑛 = 90

Activity 2: Find My t-Value! Directions: Compute the t-value of the following. Write your answer to the nearest thousandths. Show your solutions. 1. 𝑥 = 16.4

𝜇 = 15.86 𝑠 = 1.25 𝑛 = 21

4. 𝑥 = 1.83

𝜇 = 1. 27 𝑠 = 2.15 𝑛 = 10

2. 𝑥 = 246

𝜇 = 245. 85 𝑠 = 3.25 𝑛 = 29

3. 𝑥 = 9.5

𝜇 = 8.25 𝑠 = 1.45 𝑛 = 16

5. 𝑥 = 30. 18

𝜇 = 31. 23 𝑠 = 3.15 𝑛 = 23

Activity 3: Compute Me! Directions: Solve the following. Write your answer to the nearest thousandths.

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1. 𝑥 𝜇 𝜎 𝑛

2. 𝑥 = 19.8 𝑠=4 𝜇 = 18.3 𝑛 = 11

3. 𝑥 = 12.5 𝑠=3 𝜇 = 10.75 𝑛 = 18

4. 𝑥 = 125.3 𝑠=5 𝜇 = 124 𝑛 = 24

5. 𝑥 = 25.4 𝜇 = 22.6 𝜎 = 15 𝑛 = 118

6. 𝑥 = 18.1 𝑠=3 𝜇 = 19.2 𝑛 = 15

7. 𝑥 = 98.7 𝜇 = 4.6 𝜎 = 99.1 𝑛 = 105