Mathematics 215: Introduction to Statistics

Study Guide

Unit 5: Tests of Hypotheses for Two or More Populations

In Unit 4, we began a discussion of the field of statistical inference through problems involving confidence interval estimates and hypothesis tests about the mean and proportion for a single population.

In Unit 5, we extend our consideration of statistical inference to hypothesis tests involving the mean and proportion for two or more populations. We also examine other common tests of hypotheses, including tests for experiments with more than two categories, tests about contingency tables, and tests about the variance and standard deviation of a single population.

After completing Unit 5, you will be able to conduct tests of hypotheses that can apply to a wide range of real-world situations. As an example, if you are an owner of a chain of retail clothing stores, you might compare the average dollar sales per square foot of retail space achieved at each of your different store locations. Perhaps you might be part of a medical research team comparing the proportion of patients that respond positively to alternative treatments for a serious disease. As a counselor at a large university, you might need to test the effectiveness of a new examination-writing strategy on a cohort of freshman students. As a criminologist, you might want to see if there is a relation between gender and attitudes towards capital punishment.

Unit 5 of MATH 215 consists of the following sections:

5-1 Inferences About the Difference Between Two Population Means for Independent Samples: ${\sigma }_1$ and ${\sigma }_2$ Known 5-2 Inferences About the Difference Between Two Population Means for Independent Samples: ${\sigma }_1$ and ${\sigma }_2$ Unknown but Equal 5-3 Inferences About the Difference Between Two Population Means for Paired Samples 5-4 Inferences About the Difference Between Two Population Proportions for Large and Independent Samples 5-5 Goodness-of-Fit Tests 5-6 Tests for Independence and Homogeneity 5-7 Inferences About the Population Variance 5-8 Analysis of Variance

The unit also contains a self-test. When you have completed the material for this unit, including the self-test, complete Assignment 5.

Section 5-1: Inferences About the Difference Between Two Population Means for Independent Samples: ${\sigma }_1$ and ${\sigma }_2$ Known

Outcomes

After completing the readings and exercises for this section, you should be able to do the following:

1. define, and use in context, the following key terms:
   • independent samples versus dependent samples
   • sampling distribution of the difference between two sample means, ${\overline{x}}_1-{\overline{x}}_2$
2. use the critical value approach to perform a hypothesis test about the difference between two population means, $m_1-m_2$, based on independent samples, whose population standard deviations, ${\sigma }_1$ and ${\sigma }_2$, are both known.

Reading

Read the following sections in Chapter 10 of the textbook:

• Chapter 10 Introduction

• Section 10.1

  Note:

  • Omit Section 10.1.3.
  • In Section 10.1.4, omit the information about the $p\text{-value}$ approach. You are responsible for only the critical value approach.

Be prepared to read the material in Chapter 10 at least twice—the first time for a general overview of topics, and the second time to concentrate on the terms and examples presented. Return to these sections when you need to review these topics.

Supplementary Video Resources

These videos provide alternative explanations and further exploration of the concepts and techniques presented in the assigned textbook readings.

Video Related to Chapter 10

• The Sampling Distribution of the Difference in Sample Means (jbstatistics)

Videos Related to Section 10.1

• Inference for Two Means: Introduction (jbstatistics)
• Hypothesis Testing - Two Means: Large Independent Samples (mathtutordvd)
• Two Populations: $z\text{-test}$ with Hypothesis (Brandon Foltz)

Exercises

Complete the following exercises from Chapter 10 of the textbook (page numbers are for the downloadable eText):

• Exercises 10.7 c., 10.9 b., and 10.11 b. on page 403
  • Note: Use only the critical value approach for these exercises.

Show your work as you develop your answers.

Solutions are provided in the Student Solutions Manual for Chapter 10 (interactive textbook) and on pages AN14 and AN15 in the Answers to Selected Odd-Numbered Exercises (downloadable eText).

Remember, it is very important that you make a concerted effort to answer each question independently before you refer to the solutions. If your answers differ from those provided and you cannot understand why, contact your tutor for assistance.

Section 5-2: Inferences About the Difference Between Two Population Means for Independent Samples: ${\sigma }_1$ and ${\sigma }_2$ Unknown but Equal

Outcomes

After completing the readings and exercises for this section, you should be able to use the critical value approach to perform a hypothesis test about the difference between two population means, $m_1-m_2$, based on independent samples, with population standard deviations, ${\sigma }_1$ and ${\sigma }_2$, unknown but equal.

Reading

Read Section 10.2 in Chapter 10 of the textbook.

1. Note:
   • Omit Section 10.2.1.
   • In Section 10.2.2, omit the information about the $p\text{-value}$ approach. You are responsible for only the critical value approach.

Supplementary Video Resources

These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 10.2 of the textbook.

• How To ... Select the Correct $t\text{-test}$ to Compare Two Means (Eugene O’Loughlin)
• Pooled or Unpooled Variance $t$ Tests and Confidence Intervals (To Pool or Not to Pool?) (jbstatistics)
• Pooled-variance $t$ Tests and Confidence Intervals: Introduction (jbstatistics)
• Pooled-variance $t$ Tests and Confidence Intervals: an Example (jbstatistics)

Exercises

Complete the following exercises from Chapter 10 of the textbook (page numbers are for the downloadable eText):

• Exercises 10.19 b., 10.21 b., and 10.23 c. on pages 410–411.
  • Note: Use only the critical value approach for these exercises.

  Solutions are provided in the Student Solutions Manual for Chapter 10 (interactive textbook) and on page AN15 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

Section 5-3: Inferences About the Difference Between Two Population Means for Paired Samples

Outcomes

After completing the readings and exercises for this section, you should be able to do the following:

1. define, and use in context, the term “paired samples” (or “matched samples”).
2. use the critical value approach to perform hypothesis tests about the difference between two population means based on paired samples.

Reading

Read Section 10.4 in Chapter 10 of the downloadable eText.

1. Note:
   • Omit Section 10.3 entirely.
   • Omit Section 10.4.1.
   • In Section 10.4.2, omit the information about the $p\text{-value}$ approach. You are responsible for only the critical value approach.

Supplementary Video Resources

These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 10.4 of the textbook.

• Two Populations: Matched Sample $t\text{-test}$ (Brandon Foltz)
• The Paired Difference $t$ Procedure (jbstatistics)
• An Example of a Paired Difference $t$ Test and Confidence Interval (jbstatistics)

Exercises

Complete the following exercises from Chapter 10 of the textbook (page numbers are for the downloadable eText):

• Exercises 10.37 and 10.39 b. on pages 424–425.

  Note: Use only the critical value approach for these exercises.

  Solutions are provided in the Student Solutions Manual for Chapter 10 (interactive textbook) and on page AN15 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

Section 5-4: Inferences About the Difference Between Two Population Proportions for Large and Independent Samples

Outcomes

After completing the readings and exercises for this section, you should be able to do the following:

1. define, and use in context, the concept of “sampling distribution of a difference of two population proportions, $p_1$ and $p_2$ .”
2. use the critical value approach to perform hypothesis tests about the difference between two population proportions based on large and independent samples.
3. use the $p\text{-value}$ approach to perform hypothesis tests about the difference between two population proportions based on large and independent samples.

Reading

Read Section 10.5 in Chapter 10 of the textbook.

1. Note:
   • Omit Section 10.5.2.
   • In Section 10.5.3, you are responsible for both the critical value approach and the $\mathit{p}\mathbf{\text{-value}}$ approach.

Supplementary Video Resources

These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 10.5 of the textbook.

• An Introduction to Inference for Two Proportions (jbstatistics)
• Inference for Two Proportions: an Example of a Confidence Interval and a Hypothesis Test (jbstatistics)
• Inference for the Difference of Two Proportions (Bryan Nelson)

Exercises

1. Complete the following exercises from Chapter 10 of the textbook (page numbers are for the downloadable eText):

   • Exercises 10.51 c., 10.53 c., and 10.55 b. on page 432

     Note: If the question does not specify which approach to use for the hypothesis test, assume the critical value approach.

   • Supplementary Exercises 10.57 b., 10.59 b., 10.65 b., and 10.67 b. on pages 433–435

     Note: If the question does not specify which approach to use for the hypothesis test, assume the critical value approach.

2. Complete Questions 1, 3 b., and 7 b. in the Self-Review Test for Chapter 10 (page 436 of the downloadable eText).

   Note: If the question does not specify which approach to use for the hypothesis test, assume the critical value approach.

   Solutions are provided in the Student Solutions Manual for Chapter 10 (interactive textbook) and on pages AN15 and AN16 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

Optional Extra Practice

For extra practice with the material presented in this section, you can complete the following questions and exercises, for which the solutions are provided in the textbook:

1. Any odd-numbered chapter-section practice questions and Supplementary Exercises that are not assigned above
2. The odd-numbered Advanced Exercises at the end of Chapter 10 (pages 435–436 of the downloadable eText)

Section 5-5: Goodness-of-Fit Tests

Outcomes

After completing the readings and exercises for this section, you should be able to do the following:

1. define, and use in context, the following key terms:
   • chi-square distribution
   • multinomial experiment
   • observed frequency
   • expected frequency
2. use the critical value approach to perform hypothesis tests about goodness of fit.

Reading

Read the following sections in Chapter 11 of the textbook:

• Chapter 11 Introduction
• Section 11.1
• Section 11.2

Be prepared to read the material in Chapter 11 at least twice—the first time for a general overview of topics, and the second time to concentrate on the terms and examples presented. Return to these sections when you need to review these topics.

Supplementary Video Resources

These videos provide alternative explanations and further exploration of the concepts and techniques presented in the assigned textbook readings.

Videos Related to Section 11.1

• An Introduction to the Chi-square Distribution (${\chi }^2$) (jbstatistics)
• Using the Chi-square Table to Find Areas and Percentiles (jbstatistics)
• Introduction to the Chi-square Test (Brandon Foltz)

Videos Related to Section 11.2

• Chi-squared Test (Boseman Science)
• Chi-square Tests for One-way Tables (jbstatistics)
• Chi-square Tests: Goodness-of-Fit for the Binomial Distribution (jbstatistics)

Exercises

Complete the following exercises from Chapter 11 of the textbook (page numbers are for the downloadable eText):

• Exercises 11.3 and 11.5 on page 451

• Exercises 11.11,  11.15, and 11.17 on pages 458–459

  Note: In all test of hypothesis questions, use the critical value approach.

  Solutions are provided in the Student Solutions Manual for Chapter 11 (interactive textbook) and on page AN16 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

Section 5-6: Tests for Independence and Homogeneity

Outcomes

After completing the readings and exercises for this section, you should be able to do the following:

1. define the term “contingency table,” and use contingency tables to solve problems.
2. use the critical value approach to perform hypothesis tests about the independence of two attributes of a population.
3. Use the critical value approach to perform hypothesis tests about the homogeneity of two or more populations.

Reading

Read Section 11.3 in Chapter 11 of the textbook.

Supplementary Video Resources

These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 11.3 of the textbook.

• Chi-square Tests of Independence (Chi-square Tests for Two-way Tables) (jbstatistics)
• Chi-square Test – with contingency table (Math Meeting)
• How to calculate Chi-square Test for Independence (Two-way) (statisticsfun)

Exercises

Complete the following exercises from Chapter 11 of the textbook (page numbers are for the downloadable eText):

• Exercises 11.23, 11.25, 11.27, and 11.29 on pages 467–468

  Solutions are provided in the Student Solutions Manual for Chapter 11 (interactive textbook) and on page AN16 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

Section 5-7: Inferences About the Population Variance

Outcome

After completing the readings and exercises for this section, you should be able to use the critical value approach to perform a hypothesis test for the population variance, ${\sigma }^2$, or for the population standard deviation, $\sigma$.

Reading

Read Section 11.4 in Chapter 11 of the textbook:

1. Note:
   • Omit Section 11.4.1.
   • In Section 11.4.2, use the critical value approach.

Supplementary Video Resources

These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 11.4 of the textbook.

• Hypothesis Tests for the Variance (using the Chi-Square Distribution) (Brandon Foltz)
• Introduction to Inference for One Variance (assuming a Normally Distributed Population) (jbstatistics)
• Inference for a Variance: an example of a Confidence Interval and a Hypothesis Test (jbstatistics)

Exercises

1. Complete the following exercises from Chapter 11 of the textbook:
   • Exercises 11.33, 11.35 b. and 11.37 b. on page 474

2. Complete Questions 1–13 in the Self-Review Test for Chapter 11 (page 478 of the downloadable eText). Omit Question 13 a.

   Solutions are provided in the Student Solutions Manual for Chapter 11 (interactive textbook) and on pages AN16 and AN17 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

Optional Extra Practice

For extra practice with the material presented in this section, you can complete the following questions and exercises, for which the solutions are provided in the textbook:

1. Any odd-numbered chapter-section practice questions that are not assigned above
2. The odd-numbered Supplementary Exercises and Advanced Exercises at the end of Chapter 11 (pages 476–477 of the downloadable eText)

Section 5-8: Analysis of Variance

Outcomes

After completing the readings and exercises for this section, you should be able to do the following:

1. define, and use in context, the following key terms:
   • $F$ distribution
   • one-way analysis of variance (ANOVA)
2. use the critical value approach to perform a one-way ANOVA test.

Reading

Read the following sections in Chapter 12 of the textbook:

• Chapter 12 Introduction
• Section 12.1
• Section 12.2

Be prepared to read the material in Chapter 12 at least twice—the first time for a general overview of topics, and the second time to concentrate on the terms and examples presented. Return to these sections when you need to review these topics.

Supplementary Video Resources

These videos provide alternative explanations and further exploration of the concepts and techniques presented in the assigned textbook readings.

Videos Related to Section 12.1

• What is the $F\text{-Distribution}$ in Statistics? (mathtutordvd)
• Using the $F\text{-Distribution}$ Tables in Statistics (mathtutordvd)
• An Introduction to the $F$ Distribution (jbstatistics)
• Using the $F$ Table to Find Areas (jbstatistics)

Videos Related to Section 12.2

• Analysis of Variance (ANOVA) Overview in Statistics (mathtutordvd)
• ANOVA Basics – The Grand Mean (mathtutordvd)
• How to Calculate and Understand Analysis of Variance (ANOVA) $F$ Test (statisticsfun)
• ANOVA: a Visual Introduction (Brandon Foltz)
• One-way ANOVA: a Visual Tutorial (Brandon Foltz)
• One-way ANOVA: Understanding the Calculation (Brandon Foltz)

Exercises

1. Complete the following exercises from Chapter 12 of the textbook (page numbers are for the downloadable eText):
   • Exercises 12.3 and 12.5 on pages 485–486
   • Exercises 12.11, 12.13, and 12.15 on page 495

2. Complete Questions 1–10 in the Self-Review Test for Chapter 12 (page 498 of the downloadable eText).

   Solutions are provided in the Student Solutions Manual for Chapter 12 (interactive textbook) and on pages AN17 and AN18 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

3. Complete the Unit 5 Self-Test below.

Optional Extra Practice

For extra practice with the material presented in this section, you can complete the following questions and exercises, for which the solutions are provided in the textbook:

1. Any odd-numbered chapter-section practice questions that are not assigned above
2. The odd-numbered Supplementary Exercises and Advanced Exercises at the end of Chapter 12 (pages 496–498 of the downloadable eText)

Assignment 5

Once you have completed the Unit 5 Self-Test below, complete Assignment 5. You can access the assignment in the Assessment section of the course home page. Once you have completed the assignment, submit it to your tutor for marking using the drop box on the page for Assignment 5.

Unit 5 Self Test

The self-test questions are shown here for your information. Download the Unit 5 Self-Test document and write out your answers. Show all your work and keep your calculations to four decimal places, unless otherwise stated. You can access the solutions to this self-test on the course home page.

1. Circle True (T) or False (F) for each of the following:
   1. TF

      Testing an alternative hypothesis that the mean of the first population is less than the mean of the second population is the same as: $\mu 1-\mu 2<0$.
   2. TF

      When conducting a test of hypothesis involving two population means, if both random samples exceed 30 and the population standard deviations are unknown, then the standard normal distribution can be used to find the critical values.
   3. TF

      When conducting a test of hypothesis involving two population means, if both random samples exceed 30 and both population variances are unknown but equal, then the pooled standard deviation is used in the computation of the test statistic.
   4. TF

      The paired or matched sample test is appropriate to use when the two randomly selected samples are independent.
   5. TF

      In a hypothesis test for independence with a contingency table, the alternative hypothesis is that the two variables are related.
   6. TF

      A hypothesis test where the null hypothesis is that three or more population means are equal is called ANOVA.

2. Economic research shows that in any given month, the unemployment rate of college graduates is significantly lower than the unemployment rate of high-school graduates. The unemployment rate refers to the proportion of graduates registered as unemployed in any given month. In a random sample of 1,200 college graduates, 60 were unemployed; and in a random sample of 1,000 high-school graduates (no college), 64 were unemployed.

   At the 1% significance level, do the samples provide sufficient evidence to conclude that college graduates have a lower unemployment rate than high-school graduates? Use the $p\text{-value}$ approach and show all key steps.

3. A heart specialist wants to see if she can lower the cholesterol levels in 6 patients by enrolling these people in a rigorous six-month exercise program. The cholesterol levels for each of the patients before and after taking the exercise program are shown in the table below.

   At $\alpha =0.05$, did the cholesterol level decrease on average after taking the exercise program? Using the critical value approach, conduct the appropriate test of hypothesis. Show all key steps. Assume that the population of cholesterol level differences is normally distributed.

   Patient	1	2	3	4	5	6
   Before	243	216	214	222	206	219
   After	215	202	198	195	204	213

4. A government health care agency reported, based on a random sample of 16 women who have health insurance, that insured women spend on average 2.3 days in the hospital for a routine childbirth. Based on a separate independent random sample of 16 women who do not have health insurance, uninsured women reportedly spend on average 1.9 days in the hospital for routine childbirth. The standard deviation of the first sample is equal to 0.6 day, and the standard deviation of the second sample is 0.3 day.

   At $\alpha =0.01$, test the claim that the mean hospital stay is the same for insured and uninsured women. Assume both samples come from normal populations with equal variances. Show all key steps in using the critical value approach.

5. To test the effectiveness of a new drug, a pharmaceutical manufacturer randomly selected 100 patients and cross-classified the results in the following table. At $\alpha =0.10$, can the researcher conclude that the effectiveness of the drug is related to gender? Show all the key steps of the critical value approach.

   	Effective	Not effective
   Female	35	15
   Male	20	30

6. A nutritionist wishes to see whether there is any difference in the mean weight loss of individuals following one of three special diets. Individuals are randomly assigned to three groups and placed on the diet for 10 weeks. The weight losses (in kg) are shown in the table below. At $\alpha =0.05$, can the nutritionist conclude that there is a difference in the mean weight loss between the three diets? Given that all the necessary assumptions are satisfied (three normally distributed populations, etc.), use the critical value method and show all key steps.

   Protein-based diet	Carbohydrates-based diet	Fats-based diet
   2	5	3
   3	6	2
   3	5	2
   1	7	4
   	4
   	3

7. A large coffee-house chain wants to determine if its regular customers have a preference in the type of music that is played over the coffee houses’ speaker systems. A random sample of 100 regular customers is selected and the number of customers who prefer each type of music is recorded in the table below. At the 2.5% level of significance, can you reject the hypothesis that there is no difference in customer preference between the four types of music? Show all key steps for the critical value approach.

   Music type	Country	Rock	Jazz	Classical
   Number of customers	30	30	20	20

8. A random sample of the lifetimes of 24 watches manufactured under the same brand name displayed a standard deviation of 3.5 months. At the 1% level of significance, test the hypothesis that the population standard deviation of the lifetimes of this brand name of watch is less than 4 months. Show all key steps using the critical value method. Assume that the lifetimes of all the watches manufactured under this same brand name are normally distributed.