11 Statistics and Probability Module 5 Test of Hypothesis Department of Education • Republic of the Philippines 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. v Table of Contents Overview ……………………………………..……..…… 1 Module Content ………………………………………………… 1 What I Need to Know ………………………………………………… 2 General Instructions ………………………………………………… 3 What I Know ………………………………………………… 4 Lessons/Concept Lesson 1 – Basic Concepts in Hypothesis Testing……………… 6 Activity 1 ………………………………………………… 6 Activity 2 …………….………………………………..… 7 Activity 3 ……………………………………………..…. 11 Activity 4 ………………………………………………… 15 Lesson 2 – Rejection Region and Level of Significance ……… 16 Activity 1 …………….………………………………… 21 Lesson 3 – Test on Population Mean…………………………….. 23 Activity 1 ………………………………………………… 23 Activity 2 ………………………………………………… 29 Lesson 4 – Test on Population Proportion .……………………… 30 Activity 1 ………………………………………………… 32 Activity 2 ………………………………………………… 35 Assessment ………………………………………………… 36 References ………………………………………………… 47 1 Module 5 Test of Hypothesis Overview Hypothesis testing is a statistical method that is used in making statistical decisions using experimental data. Basically, it is a process of gathering evidences to either accept or reject a claim, a guess, or an assumption, known as hypothesis. In real life, we are doing hypothesis testing every time we need to make decisions on something that affect our lives. As students you need to make decisions by looking both the positive and negative sides of the problem that confronted you before making any decision. Unknowingly, your decision to enroll in the Open Senior High School went to a series of hypothesis testing. You were confronted with a lot of “what ifs” until finally you decided to be here, one of the pioneers of the Open Senior High School Program. In the previous module, you learned that statistical inference is concerned with either estimation or evaluation of a statement or claim about a parameter or a distribution. The focus of this module is on evaluation of a claim about a parameter or a distribution which is done through a statistical test of hypothesis. This module will equip you with the basic knowledge in testing hypothesis concerning population mean and population proportion which in turn will prepare you to your future journey to the world of practical research. The lessons are arranged and presented in an easier way and are supplemented with activities and exercises that would concretize your learning. So, sit back, relax and enjoy doing the module. Module Contents The lessons that we will be dealing with are the following: Lesson 1 – Basic Concepts in Hypothesis Testing Lesson 2 – Rejection Region and Level of Significance Lesson 3 – Test on Population Mean Lesson 4 – Test on Population Proportion 2 What I need to know? Once you are done with this module, you should be able to: 1. illustrate (a) null hypothesis, (b) alternative hypothesis, (c) level of significance, (d) rejection region, and (e) types of errors in hypothesis testing (M11/12SP-IVa-1). 2. calculates the probabilities of committing a Type I and Type II error (M11/12SPIVa-2). 3. identifies the parameter to be tested given a real-life problem (M11/12SP-IVa-3) 4. formulate the appropriate null and alternative hypothesis on a population mean (M11/12SP-IVb-1) 5. identify the appropriate form of the test-statistic when: (a) the population variance is assumed to be known (b) the population variance is assumed to be unknown; and (c) the Central Limit Theorem is to be used (M11/12SP-IVb-2) 6. identify the appropriate rejection region for a given level of significance when: (a) the population variance is assumed to be known (b) the population variance is assumed to be unknown (c) the Central Limit Theorem is to be used (M11/12SP-IVc-1) 7. compute for the test-statistic value of population mean (M11/12SP-IVd-1). 8. draw conclusion about the population mean based on the test-statistic value and the rejection region (M11/12SP-IVd-2). 9. solve problems involving test of hypothesis on the population mean (M11/12SPIVe-1). 10.formulate appropriate null and alternative hypotheses on a population proportion (M11/12SP-IVe-2), 11.identify the appropriate form of test statistic when the Central Limit Theorem is to be used (M11/12SP-IVe-3), 12.identify the appropriate rejection region for a given level of significance when the Central Limit Theorem is to be used (M11/12SP-IVe-4), 13.compute for the test statistic value of population proportion (M11/12SP-IVf-1), 14.draw conclusion about the population proportion based on the test-statistic value and the rejection region (M11/12SP-IVf-2), and 15. Solve problems involving test of hypothesis on the population proportion (M11/12SP-IVf-2). 3 General Instructions In order to get the most from this module and achieve its objectives, here are some TIPS for you: Read the texts and follow instructions carefully in each lesson Take note and record points for clarification. Do the activities to fully understand each lesson. Answer the self-check to monitor what you already learned in each lesson Answer the posttest Check your answer in the posttest (against the key to correction) Be aware of the following terms. Hypothesis A claim, guess, assumption, or conjecture about a population. It is the starting point of an investigation. Null Hypothesis A claim that denotes “absence” such as absence of difference, absence of relationship, or equality to a certain value. It is denoted by Ho. Alternative Hypothesis A claim that denotes “presence” such as presence of difference, presence of relationship, or inequality to a certain value. It is denoted by Ha. Type I error When we reject the null hypothesis, although that hypothesis was true. Type I error is denoted by alpha (𝛼𝛼). In hypothesis testing, the normal curve that shows the critical region is called the alpha (𝛼𝛼)region Type II error When we accept the null hypothesis but it is false. Type II errors are denoted by beta (𝛽𝛽). In Hypothesis testing, the normal curve that shows the acceptance region is called the beta (𝛽𝛽) region Power Usually known as the probability of correctly accepting the null hypothesis. 1 – beta is called power of the analysis. Level of Significance Refers to the degree of significance in which we accept or reject the null-hypothesis. One-tailed test When the given statistical hypothesis is one value like H0: μ1 = μ2, it is called the one-tailed test. Two-tailed test When the given statistical hypothesis assumes a less than or greater than value, it is called the two-tailed test. 4 What I Know Directions: Select the letter of the option that correctly answers the questions or completes the statement. 1. This refers to an intelligent guess, an assumption, or a claim about a population parameter which may either be true or false. A. Hypothesis C. Decision B. Test statistic D. Interpretation 2. It is a rule or method that leads to decision to accept or reject the hypothesis when sample values are gathered from the population under study. A. Estimation C. Hypothesis Testing B. Hypothesis D. Test Statistic 3. What is the purpose of hypothesis testing? A. To collect sample data and use them to formulate hypotheses about a population. B. To draw conclusions about population and then collect sample data to support the conclusions. C. To draw conclusions about populations from sample data. D. To draw conclusions about the known value of population parameter 4. What mathematical model is appropriate for decision-making about population proportion? A. Graphical representation C. z – statistic B. Normal curve D. None of these 5. The probability of rejecting the null hypothesis when it is true is called, A. Level of Confidence C. Power of the test B. Level of Significance D. Estimate interval 6. Which of the following statements is true when the null hypothesis is rejected? A. There is significant difference between parameters being compared. B. There is no significant difference between parameters being compared. C. The conclusion is guaranteed. D. The conclusion is not guaranteed. 7. What is the value of α for the 95% confidence level of a two-tailed test? A. 0.01 C. 0.10 B. 0.05 D. 0.025 8. If in the z-test of proportions, the computed z is found on the rejection region then this means that: 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 9. If p = 0.3, = 0.4, n = 50 what is the value of z? A. 0.45 C. 1.54 B. 0.63 D. 0.55 10.Under the normal curve, the middle part represents A. Confidence level C. Acceptance region B. Confidence interval D. All of