Stat Prob 11 Q4 Mod1 Tests of Hypothesis Version 2 PDF

Preview

Summary

STATISTICS & PROBABILITYQuarter 4- Module 1:Tests of HypothesisDepartment of Education ● Republic of the PhilippinesStatistics & Probability – Grade 11 Alternative Delivery Mode Quarter 4– Module 4: Tests of Hypothesis First Edition, 2020Republic Act 8293, section 176 states that: “No copyri...

Description

STATISTICS & PROBABILITY Quarter 4- Module 1: Tests of Hypothesis

Department of Education ● Republic of the Philippines

Statistics & Probability – Grade 11 Alternative Delivery Mode Quarter 4– Module 4: Tests of Hypothesis 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 included in this module are owned by their respective copyright holders. Effort has been exerted to locate and seek permission to use these materials from the respective copyright owners. The publisher and author do not represent nor claim ownership over them.

Published by the Department of Education Secretary: Undersecretary: Assistant Secretary:

Development Team of the Module Authors: Flordeliz D. Laput, MSMath, Kenny John L. Aguilar Editor: Glenn C. Arandilla Reviewers: Illustrator: Layout Artist: Management Team: Nelson B. Absin

Printed in the Philippines by: _____________________________ Department of Education – Bureau of Learning Resources (DepEd – BLR) Office Address:

______________________________________

Telefax:

______________________________________

E-mail Address:

______________________________________

Statistics & Probability Quarter 4 – Module 1 Tests of Hypothesis

This instructional material was collaboratively developed and reviewed by educators from public and private schools, colleges, and or/universities. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Department of Education at [email protected]. We value your feedback and recommendations.

Department of Education • Republic of the Philippines TABLE OF CONTENTS Cover Page Copyright Page Title Page Table of Contents Module Overview

Competency 1 Illustrating Some Key Concepts in Hypothesis Testing What’s New 1 What I Need To Know 1 What I Know 2 What’s New Lesson 1.1 Null Hypothesis 3 What Is It? Activity 1 4 What’s New Lesson 1.2 Alternative Hypothesis 5 What Is It? Activity 2 7 What’s New Lesson 1.3 Level of Significance 8 What Is It? Activity 3 9 What’s New Lesson 1.4 Critical Region 10 What Is It? Activity 4 12 What’s New Lesson 1.5 Types of Error in Hypothesis Testing 13 What Is It? Activity 5 14 What I Have Learned 15 Assessment 16 Answer Key 17 References 18 Competency 2 Illustrate Some Basic Concept in Hypothesis Testing Competency 3 Formulate the Appropriate Null and Alternative Hypotheses on a Population Mean 19 What’s New 20 What I Need To Know 20 What I Know 21 What’s New Lesson 2.1 z-Test on the Comparison Between the Population Mean and Sample Mean 21 What Is It? Activity 6 23 What’s New Lesson 2.2 t-Test on the Comparison Between the Population Mean and Sample Mean 24 What Is It? Activity 7 26 What I Have Learned 27 Assessment 27 Answer Key 28 References 29 Competency 4 Identifies the Appropriate Form of the test-statistic Competency 5 Identifies the Appropriate Rejection Region for a Given Level of Significance 30 What’s New 31 What I Need To Know 31 What I Know 32 What’s New Lesson 3.1 The Test Statistic When the Population Variance is Assumed to be Known 33 What Is It? Activity 8 34 What’s New Lesson 3.2 The Test Statistic When the Population Variance is Assumed to be Unknown 34 What Is It? Activity 9 35 What’s New Lesson 3.3 Test Statistic Using Central Limit Theorem 35 What Is It? Activity 10 37 What’s New Lesson 4.1 The Rejection Region When the Population Variance is Assumed to be Known 38 What Is It? Activity 11 41 What’s New Lesson 4.2 The Rejection Region When the Population Variance is Assumed to be Unknown 41

What Is It? Activity 12 44 What’s New Lesson 3.3 Rejection Using Central Limit Theorem 45 What Is It? Activity 13 46 What I Have Learned 47 Assessment 47 Answer Key 49 References 51 Competency 6 Draws Conclusion about the Population Mean Based on the the Test-Statistic Value and Rejection Region Competency 7 Solves Problems Involving Test of Hypothesis on the Population Mean 53 What’s New 54 What I Need To Know 54 What I Know 54 What’s New Lesson 5 Problems Involving Test of Hypothesis on the Population Mean 55 Assessment 58 Answer Key 59 References 60 Module Writer’s Profile 61 Competency 8 – 13 Tests of Hypothesis on Population Proportion 62 What I Need To Know 63 What I Know 63 What’s In 63 What is it 64 What’s More 65 What I Have Learned 65 What I Can Do 66 Assessment 68 Additional Activities 69 Module Writer’s Profile 70 Back Outside Cover

OVERVIEW This module is made for you Grade 11 learners. It is crafted in a simple and direct manner to correspond to your 21st century skills. Examples were given to support the discussion and for illustration. It was made to enhance distant learning delivery of education. We envisioned to provide an alternative way in continuing education, and to provide free, interactive and quality learning materials to you our dear learners which focused on the most essential learning competencies.

Further, this module is for Grade 11 students enrolled in Statistics and Probability subject. Inside this module, you will be asked to read and understand some terminologies, ideas, process and computations. You will also be asked to identify population parameters, illustrate null and alternative hypotheses, level of significance, critical region, and types of errors in testing hypothesis, identifies appropriate test statistic when a parameter is known, unknown and using Central Limit Theorem, formulate appropriate null and alternative hypotheses, compute critical values, construct or sketch the critical and acceptance region, draw conclusion about the population based on the test statistic and rejection region, and solve real-world problems .

I hope that this module will be an important tool to enrich your knowledge, strengthening your statistical skills and lessen your computation anxiety. God bless learners! Flordeliz Dumaran Laput Kenny John L. Aguilar

STATISTICS & PROBABILITY Quarter 4 - Module 1 Tests of Hypothesis

Learning Competency 1: Illustrates (a) Null Hypothesis (b) Alternative Hypothesis (c) Level of Significance (d) Rejection Region (e) Types of Errors in Hypothesis Testing

-

M11/12SP-IVa-1.1 M11/12SP-IVa-1.2 M11/12SP-IVa-1.3 M11/12SP-IVa-1.4 M11/12SP-IVa-1.5

2nd Semester 4th Quarter Number of Hours: 2

WHAT’S NEW LESSON 1 Some Key Concepts of Tests of Hypothesis This part of the module discusses about some key concepts of tests of hypothesis. It includes the null and alternative hypotheses, level of significance, rejection region, and the types of errors in hypothesis testing. There are activities following every discussion which were designed to test your understanding about the discussion.

Hypothesis testing is a decision-making process of evaluating claims about a population based on the characteristic of a sample from that population. It decides whether to reject or accept the null hypothesis. Some uses the following decisions: the null hypothesis is rejected or failed to reject the null hypothesis. Acceptance implies that the null hypothesis is true. Failure to reject implies that the data are not sufficient enough to reject the null hypothesis. In this module, your decision either reject the null hypothesis or the data are not sufficient enough to reject the null hypothesis. Testing hypothesis follows the following steps below. 1. 2. 3. 4. 5. 6. 7.

Formulate the null and alternative hypotheses Select the level of significance and the test type Determine test statistic to be used Compute the test statistic and critical values Construct acceptance/ rejection regions Make a decision/ draw a conclusion based on steps 4 and 5 Interpretation

What I Need to Know By the end of this lesson, you are expected to: 1. Illustrate the following:

(a) null hypothesis (b) alternative hypothesis (c) level of significance (d) rejection region (e) types of errors in hypothesis testing

To achieve the objectives of this 1module, do the following tasks:  Take your time reading the lessons carefully.  Follow the directions and/ or instructions in the activities and exercises diligently.  Answer all the given tests and exercises.

IMPORTANT REMINDER: DO NOT WRITE ANYTHING IN THIS MODULE. USE A SEPARATE NOTEBOOK WHERE YOU CAN WRITE YOUR ANSWERS FOR THIS MODULE. 9 What I know A. Before the start of the lesson proper, answer first the questions below. Read the it carefully and write the letter of the best answer in your notebook.

1. It is a process in making decisions in evaluating claims about a population. A. Null hypothesis C. Test statistic B. Alternative hypothesis D. Hypothesis testing 2. It is tested by a statistical test. A. Null hypothesis B. Alternative hypothesis

C. Test statistic D. Hypothesis testing

3. It is chosen when the parameter is larger than or smaller than the value of the null hypothesis. A. two-sided test C. Parametric test B. one-sided test D. Non-parametric test 4. This assumes that there is change, difference, relationship, or the independent variable has an effect on the dependent variable. A. Null hypothesis C. Test statistic B. Alternative hypothesis D. Hypothesis testing 5. This assumes that there is no change, no difference, no relationship, or the independent variable has no effect on the dependent variable. A. Null hypothesis C. Test statistic B. Alternative hypothesis D. Hypothesis testing 6. Null and alternative hypotheses are statements about: 2 A. population parameters. C. sample parameters. B. sample statistics. D. it depends - sometimes population parameters and sometimes sample statistics.

B. In your notebook, write nine examples of hypothesis.

WHAT’S NEW LESSON 1.1 NULL HYPOTHESIS

 Null was defined as having no value or amounting to nothing  In statistics, a hypothesis is an assumption or conjecture about a population parameter which may or may not be true.  In the scientific method, the hypothesis is constructed before any applicable research has been done.

 Null hypothesis is a statement denoted by

H 0 , that states that there is no difference,

no changes, nothing happened, no relationship between a parameter and a specific value, or the independent variable has no effect on the dependent variable.  It makes a statement about the population not the sample. The true value of the population parameter is specified in writing the null hypothesis.

In symbol, it is written using the format below H 0 : μ=100 . Null hypothesis Population Parameter

Equality Symbol

Value of the population parameter

3 WHAT IS IT

ACTIVITY 1 A. Read and understand the given statements below and find out whether it is a null hypothesis. In your notebook, write H 0 if the given is a null hypothesis. Otherwise, just leave it blank. In 2015, it was recorded that around 34% of the population in 2015 were not married. A researcher surveyed a random sample of 500 couples. He found out that 18% of them were living together but unmarried. Test at 5% significance level if the current percentage of unmarried couples is different from 34%. 1. The current percentage of unmarried couples is different from 34%. 2. The current percentage of unmarried couples is 34%. An average construction worker hourly rate pay in the Philippines is Php 62.50 with a standard deviation of Php 6.01. A random sample of 20 manufacturing workers were asked on their hourly rate and found that they had an average of Php 50 hourly rate pay with a standard deviation of Php 5.00. Construct a 90% confidence interval for the difference between the average hourly rate for construction workers and the average hourly rate for manufacturing workers. 3. There is a significant difference between the average hourly rate for construction workers and the average hourly rate for manufacturing workers. 4. There is no significant difference between the average hourly rate for construction workers and the average hourly rate for manufacturing workers. A chemist invented an additive to increase the lifespan of rechargeable battery. The said additive will extend on average the battery’s lifespan to 48 months. 5. The average lifespan extension of rechargeable battery is 48 months. 6. The average lifespan extension of rechargeable battery is not 48 months

B. Comprehension Check Questions: Write the letter of the best answer among the choices below in your notebook. 1. They worked in producing goods. A. Construction B. Manufacturing C. Chemist D. Investors 2.They worked with the composition, structure, and properties of substances and with the transformations that they undergo A. Construction B. Manufacturing C. Chemist D. Investors 3. It is square root of variance. A. Confidence Interval B. standard deviation C. Range D. Mean 4 4. It is the sum of values divided by the number of values being summed. A. Confidence Interval B. standard deviation C. Range D. Mean 5. It is a range of numbers containing possible values for the population parameter. A. Confidence Interval B. standard deviation C. Range D. Mean C. Read and understand the statement below. In your notebook, write the mathematical symbol of the null hypothesis of the following statements. 1. The average number of years to finish basic education is 14 years. 2. At least 40% of private school students transferred to public school during the COVID 19 pandemic. 3. The mean weekly expenses of a family during the COVID 19 pandemic increased at most by 15%. 4. Thirty-five percent of senior high school students enrolled to a track/ strand because of peer pressure. 5. During the COVID 19 pandemic, 8% of COVID 19 cases in the country were confirmed death cases. 6. The mean number of new normal learning facilities a school have is not more than five. 7. During the COVID 19 pandemic, more than half of the residents in cities decided to stay in their remote provinces. 8. Forty-five percent of the students attended online learning delivery mode. 9. At most, 55% of the public-school teachers were advised to stay at home during the opening of classes. 10. At least 70% of the public school used blended learning delivery mode.

WHAT’S NEW LESSON 1.2 ALTERNATIVE HYPOTHESIS

 Alternative hypothesis is a statement denoted by H 1 , is a statement that states that there is a difference, an effect, change, relationship between a parameter and a specific value, the independent variable has an effect on the dependent variable, or something happened.

 An alternative hypothesis is a statement that directly contradicts a null hypothesis by stating that that the actual value of a population parameter is less than, greater than, or not equal to the value stated in the null hypothesis.

In symbol, it is written as:

H 1 :5μ ≠ 100

H 1 : μ100

The alternative hypothesis will also determine the type of hypothesis testing will be conducted. One-tailed test will be used when using ¿ or ¿ . Two-tailed test will be used when ≠ is used. Below are the common phrases used in hypothesis testing that will guide you the correct symbol to be used in formulating alternative hypothesis. 13

greater than

less than

above

below

not equal

equal to

6

higher than

lower than

longer than

smaller than

bigger than

shorter than

increased

decreased or reduced from

different from

the same as

changed from

not changed from

not the same as e as

is

WHAT IS IT? ACTIVITY 2

A. Read and understand the given statements below. In your notebook, write H 1 if the given statement in every number is an alternative hypothesis. Otherwise, just leave it blank. In 2015, it was recorded that around 34% of the population in 2015 were not married. A researcher surveyed a random sample of 500 couples. He found that 18% of them were living together but unmarried. Test at 5% significance level if the current percentage of unmarried couples is different from 34%. 1. The current percentage of unmarried couples is different from 34%. 2. The current percentage of unmarried couples is 34%. An average construction worker hourly rate pay in the Philippines is Php 62.50 with a standard deviation of Php 6.01. A random sample of 20 manufacturing workers were asked on their hourly rate and found that they had an average of Php 50 hourly rate pay with a standard deviation of Php 5.00. Construct a 90% confidence interval for the difference between the average hourly rate for construction workers and the average hourly rate for manufacturing workers. 3. There is a significant difference between the average hourly rate for construction workers and the average hourly rate for manufacturing workers. 4. There is no significant difference between the average hourly rate for construction workers and the average hourly rate for manufacturing workers. A chemist invented an additive to increase the lifespan of rechargeable battery. The said additive will extend on average the battery’s lifespan to 48 months. 5. The average lifespan extension of rechargeable battery is not 48 months. 6. The average lifespan extension of rechargeable battery is 48 months. B. Read and understand the statements below. In your notebook, write the mathematical symbol of the alternative hypothesis of the given statement. 1. The average number of years to finish basic education is not 14 years. 2. At least 40% of private students transferred to public schools during the COVID 19 pandemic. 3. The mean expenses of a family during the COVID 19 pandemic increased at most by 15%. 4. Thirty-five percent of senior high school students enrolled to a track/ strand because of peer pressure. 5. During the COVID 19 pandemic, 8% of COVID 19 cases in the country were confirmed death cases. 7 6. The mean number of new normal learning facilities a school has is not more than five. 7. During the COVID 19 pandemic, more than half of the residents in cities decided to stay in their remote provinces. 8. Forty-five percent of the students attended online learning delivery mode. 9. At most, 55% of the public-school teachers were advised to stay at home during the opening of classes. 10. At least 70% of the public schools used blended learning delivery mode.

WHAT’S NEW LESSON 1.3 LEVEL OF SIGNIFICANCE

 Significance is defined as the quality of being statistically significant

 Level of significance, or significance level, refers to a criterion of judgment upon which a decision is made regarding the value stated in a null hypothesis. Its value is between 0 to 1 or between 0% to 100%.  The level of significance, denoted by the Greek letter alpha α , is a probability of rejecting a true null hypothesis. In public health research, alpha is usually 0.01 or 1%. In social science, alpha α is usually 0.05 or 5% and 0.10 or 10% in other studies. This implies that there is 1%, 5%, or 10% probability of rejecting a true null hypothesis. Further, it implies that the result has 99%, 95%, or 90% chance of being true, respectively.

In symbol, it is written as: α=0.01 α =0.05 α=0.10 Furthermore, if the alternative hypothesis used then alpha will be divided by 2, i.e.,

or

≠ , α =0.005 2 α =0.025 2

or

α =0.05 2

WHAT IS IT?

8

ACTIVITY 3 α 2 based on the alternative hypothesis in decimal form. Write your answer in mathematical symbol ...