math 215 c10 - study guide: unit 5
TRANSCRIPT
MATH 215 C10 - Study Guide: Unit 5In 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
5-2
5-3
5-4
5-5
5-6
5-7
5-8
The unit also contains a self-test. When you have completed the material for this unit, including the self-test, complete
Assignment 5.
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,
2. use the critical value approach to perform a hypothesis test about the difference between two population means, , based on independent samples, whose population standard deviations, and , are both known.
MATH 215 C10 - Study Guide: Unit 5
1
Read the following sections in Chapter 10 of the textbook:
Chapter 10 Introduction
Omit Section 10.1.3.
In Section 10.1.4, omit the information about the 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.
These videos provide alternative explanations and further exploration of the concepts and techniques presented in the assigned
textbook readings.
Hypothesis Testing - Two Means: Large Independent Samples (https://www.youtube.com/watch?v=KTFm7El1NBs) (mathtutordvd)
Two Populations: with Hypothesis (https://www.youtube.com/watch?v=5NcMFlrnYp8) (Brandon Foltz)
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.
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, , based on independent samples, with population
standard deviations, and , unknown but equal.
MATH 215 C10 - Study Guide: Unit 5
2
Note:
Omit Section 10.2.1.
In Section 10.2.2, omit the information about the approach. You are responsible for only the critical value approach.
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 to Compare Two Means (https://www.youtube.com/watch?v=uwTYD60nng4) (Eugene O’Loughlin)
Pooled or Unpooled Variance Tests and Confidence Intervals (To Pool or Not to Pool?) (https://www.youtube.com /watch?v=7GXnzQ2CX58) (jbstatistics)
Pooled-variance Tests and Confidence Intervals: Introduction (https://www.youtube.com/watch?v=NaZBdj0nCzQ) (jbstatistics)
Pooled-variance Tests and Confidence Intervals: an Example (https://www.youtube.com/watch?v=Q526z1mz4Sc) (jbstatistics)
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).
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.
Read Section 10.4 in Chapter 10 of the downloadable eText.
Note:
3
In Section 10.4.2, omit the information about the approach. You are responsible for only the critical value approach.
These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 10.4
of the textbook.
An Example of a Paired Difference Test and Confidence Interval (https://www.youtube.com/watch?v=upc4zN_-YFM) (jbstatistics)
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).
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, and .”
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 approach to perform hypothesis tests about the difference between two population proportions based on large and independent samples.
Read Section 10.5 in Chapter 5 of the textbook.
Note:
Omit Section 10.5.2.
In Section 10.5.3, you are responsible for both the critical value approach and the approach.
These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 10.5
of the textbook.
4
Inference for Two Proportions: an Example of a Confidence Interval and a Hypothesis Test (https://www.youtube.com /watch?v=OIYkOiQX3fk&index=3&list=UUiHi6xXLzi9FMr9B0zgoHqA) (jbstatistics)
Inference for the Difference of Two Proportions (https://www.youtube.com/watch?v=t1fRwSGgq2A) (Bryan Nelson)
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).
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)
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
5
expected frequency
2. use the critical value approach to perform hypothesis tests about goodness of fit.
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.
These videos provide alternative explanations and further exploration of the concepts and techniques presented in the assigned
textbook readings.
Using the Chi-square Table to Find Areas and Percentiles (https://www.youtube.com/watch?v=C-0uN1inmcc) (jbstatistics)
Introduction to the Chi-square Test (https://www.youtube.com/watch?v=SvKv375sacA) (Brandon Foltz)
Chi-squared Test (https://www.youtube.com/watch?v=WXPBoFDqNVk) (Boseman Science)
Chi-square Tests for One-way Tables (https://www.youtube.com/watch?v=gkgyg-eR0TQ) (jbstatistics)
Chi-square Tests: Goodness-of-Fit for the Binomial Distribution (https://www.youtube.com/watch?v=O7wy6iBFdE8) (jbstatistics)
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).
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.
MATH 215 C10 - Study Guide: Unit 5
6
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.
Read Section 11.3 in Chapter 11 of the textbook.
These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 11.3 of
the textbook.
How to calculate Chi-square Test for Independence (Two-way) (https://www.youtube.com/watch?v=xEiQn6sGM20) (statisticsfun)
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).
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, , or for the population standard deviation, .
Read Section 11.4 in Chapter 11 of the textbook:
Note:
These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 11.4 of
the textbook.
Introduction to Inference for One Variance (assuming a Normally Distributed Population) (https://www.youtube.com /watch?v=lyd4V8DFCjM&list=UUiHi6xXLzi9FMr9B0zgoHqA&index=2) (jbstatistics)
Inference for a Variance: an example of a Confidence Interval and a Hypothesis Test (https://www.youtube.com
MATH 215 C10 - Study Guide: Unit 5
7
/watch?v=tsLGbpu_NPk&list=UUiHi6xXLzi9FMr9B0zgoHqA) (jbstatistics)
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).
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)
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:
distribution
one-way analysis of variance (ANOVA)
2. use the critical value approach to perform a one-way ANOVA test.
Read the following sections in Chapter 12 of the textbook:
Chapter 12 Introduction
8
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.
These videos provide alternative explanations and further exploration of the concepts and techniques presented in the assigned
textbook readings.
An Introduction to the Distribution (https://www.youtube.com/watch?v=G_RDxAZJ-ug) (jbstatistics)
Using the Table to Find Areas (https://www.youtube.com/watch?v=mSn55vREkIw) (jbstatistics)
Analysis of Variance (ANOVA) Overview in Statistics (https://www.youtube.com/watch?v=CS_BKChyPuc) (mathtutordvd)
ANOVA Basics – The Grand Mean (https://www.youtube.com/watch?v=qSz9xfnXSwg) (mathtutordvd)
How to Calculate and Understand Analysis of Variance (ANOVA) Test (https://www.youtube.com/watch?v=- yQb_ZJnFXw&nohtml5=False) (statisticsfun)
ANOVA: a Visual Introduction (https://www.youtube.com/watch?v=0Vj2V2qRU10) (Brandon Foltz)
One-way ANOVA: a Visual Tutorial (https://www.youtube.com/watch?v=JgMFhKi6f6Y) (Brandon Foltz)
One-way ANOVA: Understanding the Calculation (https://www.youtube.com/watch?v=UrRYITjDOww) (Brandon Foltz)
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.
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
MATH 215 C10 - Study Guide: Unit 5
9
2. The odd-numbered Supplementary Exercises and Advanced Exercises at the end of Chapter 12 (pages 496–498 of the downloadable eText)
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.
The self-test questions are shown here for your information. Download the Unit 5 Self-Test (https://fst-course.athabascau.ca
/science/math/215/r10/self_test/self_test05.html) 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:
a. T F Testing an alternative hypothesis that the mean of the first population is less than the mean of the second population is the same as: .
b. T F 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.
c. T F 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.
d. T F The paired or matched sample test is appropriate to use when the two randomly selected samples are independent.
e. T F In a hypothesis test for independence with a contingency table, the alternative hypothesis is that the two variables are related.
f. T F 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 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 , 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.
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
MATH 215 C10 - Study Guide: Unit 5
10
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 , 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 , can the researcher conclude that the effectiveness of the drug is related to gender? Show all the key steps of the critical value approach.
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 , 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.
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.
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.
MATH 215 C10 - Study Guide: Unit 5
11
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:
a. T F Testing an alternative hypothesis that the mean of the first population is less than the mean of the second population is the same as: .
b. T F 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.
c. T F 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.
d. T F The paired or matched sample test is appropriate to use when the two randomly selected samples are independent.
e. T F In a hypothesis test for independence with a contingency table, the alternative hypothesis is that the two variables are related.
f. T F 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 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 , 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.
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.
MATH 215 C10 - Self-Test: Unit 5
1
At , 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 , can the researcher conclude that the effectiveness of the drug is related to gender? Show all the key steps of the critical value approach.
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 , 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.
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.
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.
MATH 215 C10 - Self-Test: Unit 5
2
Show all your work and keep your calculations to four decimal places, unless otherwise stated.
1. Circle True (T) or False (F) for each of the following:
a. F Testing an alternative hypothesis that the mean of the first population is less than the mean of the second population is the same as: .
b. T 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.
c. F 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.
d. T The paired or matched sample test is appropriate to use when the two randomly selected samples are independent.
e. F In a hypothesis test for independence with a contingency table, the alternative hypothesis is that the two variables are related.
f. F 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 approach and show all key steps.
Solution:
Given and ,
the sample proportion of college graduates unemployed is , and the sample proportion of high-
school graduates unemployed is .
Step 1: State the null hypothesis ( ) and the alternative hypothesis ( )
: ( is the same as )
Step 2: Select the distribution to use.
Select the distribution as the sample sizes are large and the sample sizes multiplied by the sample proportions exceed 5.
Step 3: Calculate the .
T
F
T
F
T
T
-value
-value
1
Step 4: Make a decision.
Since the of 0.0793 exceeds , do not reject .
We cannot conclude that the proportion of college graduates who are unemployed is significantly less than the proportion of high-school graduates who are unemployed.
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 , 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.
Solution:
This is a mean of paired samples problem, where the two samples are dependent.
Let = (cholesterol level after program) – (cholesterol level before program)
To find , set up the following table:
observed
observed
-value
-value
and
2
The hypothesis test:
Step 1: State the null hypothesis ( ) and the alternative hypothesis ( ). Test: (After Before) or (After – Before) , so:
: (cholesterol level did not decrease)
: (cholesterol level decreased)
Since the population is normally distributed with the population standard deviation unknown, the distribution is used.
Step 3: Determine the rejection and non-rejection regions.
Since contains “ ”, this is a one-tailed test with the left tail having an area of .
The .
Step 5: Make a decision.
Since the falls in the rejection region, we reject . Conclusion: The mean cholesterol level has decreased
after taking the exercise program.
observed
observed
3
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 , 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.
Solution:
Step 1: State the null hypothesis ( ) and the alternative hypothesis ( ).
: (mean hospital stay is the same for both populations)
: (mean hospital stay differs between both populations)
Step 2: Select the distribution to use.
Since the populations are normally distributed with population standard deviations unknown, the distribution is used.
Step 3: Determine the rejection and non-rejection regions.
Since contains “ ”, this is a two-tailed test with the combined areas of both rejection regions equal to 0.01. The .
Step 4: Calculate the value of the test statistic.
, where and , where , assuming is true.observed
MATH 215 C10 - Self-Test Answer Key: Unit 5
4
Step 5: Make a decision.
Since falls in the non-rejection region, do not reject . We cannot conclude that there is a
significant difference in the mean hospital stay of insured women versus uninsured women.
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 , can the researcher conclude that the effectiveness of the drug is related to gender? Show all the key steps of the critical value approach.
Solution:
Step 1: State the null hypothesis ( ) and alternative hypothesis ( ).
: The effectiveness of the drug is independent of gender.
: The effectiveness of the drug is related to gender.
Step 2: Select the distribution for tests involving the relation between two categorical variables.
Step 3: Determine the rejection and non-rejection regions.
Since this is a test for independence, it is a right -tailed test with with , where = the number of rows and = the number of columns in
the contingency table given above.
= sample size =
Step 4: Calculate the value of the test statistic.
5
Reject as falls in the rejection region.
Conclusion: The effectiveness of the drug is related to gender.
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 , 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.
Solution:
To test the difference in the mean weight loss between diets, follow the steps below.
total weight loss across all three diets
Between-samples sum of squares
Within-samples sum of squares
:
row total × column total
MATH 215 C10 - Self-Test Answer Key: Unit 5
6
Step 2: Select the distribution to use.
Because we are comparing the means of three normally distributed populations, we use the distribution.
Step 3: Determine the rejection and non-rejection regions.
Since this is a one-way ANOVA test, it is a right-tailed test with with and .
Step 4: Calculate the value of the test statistic.
Step 5: Make a decision.
Since falls in the rejection region, we reject . There is a difference in the mean weight loss between the
three diets.
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.
Solution:
This random sample may be classified as a multinomial experiment. To test whether the observed frequencies for an experiment follow a certain pattern or theoretical distribution, you use a goodness-of-fit test. Use the critical value approach.
To test the hypothesis, follow the steps below.
Step 1: State the null hypothesis ( ) and the alternative hypothesis ( ).
: There is no difference in customer preference between the four types of music.
: There is a difference in customer preference between the four types of music.
numerator denominator
observed MSB
7
Select the distribution for goodness-of-fit tests.
Step 3: Determine the rejection and non-rejection regions.
Since this is a goodness-of-fit test, it is a right-tailed test with with , where = number of categories of music.
Step 4: Calculate the value of the test statistic.
is the probability that a random customer’s preference of music choice is in a given category.
= sample size = customers
determined as follows:
Do not reject as falls in the non-rejection region.
Conclusion: We cannot conclude that there is a difference in customer preference between the four types of music.
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.
Solution:
:
:
Step 2: Select the distribution to use.
Since the population is normally distributed, we use the Chi-square distribution to test a hypothesis about the population variance.
Step 3: Determine the rejection and non-rejection regions.
value
observed
observed
8
© Athabasca University
Since contains “ ”, the rejection region is on the left side of the chi-square distribution, with an area of and .
Step 4: Calculate the value of the test statistic.
Step 5: Make a decision.
Since the test statistic falls in the non-rejection, we do not reject . There is no evidence that the standard deviation of the watch lifetimes is less than months.
observed
9
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
5-2
5-3
5-4
5-5
5-6
5-7
5-8
The unit also contains a self-test. When you have completed the material for this unit, including the self-test, complete
Assignment 5.
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,
2. use the critical value approach to perform a hypothesis test about the difference between two population means, , based on independent samples, whose population standard deviations, and , are both known.
MATH 215 C10 - Study Guide: Unit 5
1
Read the following sections in Chapter 10 of the textbook:
Chapter 10 Introduction
Omit Section 10.1.3.
In Section 10.1.4, omit the information about the 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.
These videos provide alternative explanations and further exploration of the concepts and techniques presented in the assigned
textbook readings.
Hypothesis Testing - Two Means: Large Independent Samples (https://www.youtube.com/watch?v=KTFm7El1NBs) (mathtutordvd)
Two Populations: with Hypothesis (https://www.youtube.com/watch?v=5NcMFlrnYp8) (Brandon Foltz)
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.
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, , based on independent samples, with population
standard deviations, and , unknown but equal.
MATH 215 C10 - Study Guide: Unit 5
2
Note:
Omit Section 10.2.1.
In Section 10.2.2, omit the information about the approach. You are responsible for only the critical value approach.
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 to Compare Two Means (https://www.youtube.com/watch?v=uwTYD60nng4) (Eugene O’Loughlin)
Pooled or Unpooled Variance Tests and Confidence Intervals (To Pool or Not to Pool?) (https://www.youtube.com /watch?v=7GXnzQ2CX58) (jbstatistics)
Pooled-variance Tests and Confidence Intervals: Introduction (https://www.youtube.com/watch?v=NaZBdj0nCzQ) (jbstatistics)
Pooled-variance Tests and Confidence Intervals: an Example (https://www.youtube.com/watch?v=Q526z1mz4Sc) (jbstatistics)
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).
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.
Read Section 10.4 in Chapter 10 of the downloadable eText.
Note:
3
In Section 10.4.2, omit the information about the approach. You are responsible for only the critical value approach.
These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 10.4
of the textbook.
An Example of a Paired Difference Test and Confidence Interval (https://www.youtube.com/watch?v=upc4zN_-YFM) (jbstatistics)
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).
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, and .”
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 approach to perform hypothesis tests about the difference between two population proportions based on large and independent samples.
Read Section 10.5 in Chapter 5 of the textbook.
Note:
Omit Section 10.5.2.
In Section 10.5.3, you are responsible for both the critical value approach and the approach.
These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 10.5
of the textbook.
4
Inference for Two Proportions: an Example of a Confidence Interval and a Hypothesis Test (https://www.youtube.com /watch?v=OIYkOiQX3fk&index=3&list=UUiHi6xXLzi9FMr9B0zgoHqA) (jbstatistics)
Inference for the Difference of Two Proportions (https://www.youtube.com/watch?v=t1fRwSGgq2A) (Bryan Nelson)
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).
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)
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
5
expected frequency
2. use the critical value approach to perform hypothesis tests about goodness of fit.
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.
These videos provide alternative explanations and further exploration of the concepts and techniques presented in the assigned
textbook readings.
Using the Chi-square Table to Find Areas and Percentiles (https://www.youtube.com/watch?v=C-0uN1inmcc) (jbstatistics)
Introduction to the Chi-square Test (https://www.youtube.com/watch?v=SvKv375sacA) (Brandon Foltz)
Chi-squared Test (https://www.youtube.com/watch?v=WXPBoFDqNVk) (Boseman Science)
Chi-square Tests for One-way Tables (https://www.youtube.com/watch?v=gkgyg-eR0TQ) (jbstatistics)
Chi-square Tests: Goodness-of-Fit for the Binomial Distribution (https://www.youtube.com/watch?v=O7wy6iBFdE8) (jbstatistics)
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).
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.
MATH 215 C10 - Study Guide: Unit 5
6
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.
Read Section 11.3 in Chapter 11 of the textbook.
These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 11.3 of
the textbook.
How to calculate Chi-square Test for Independence (Two-way) (https://www.youtube.com/watch?v=xEiQn6sGM20) (statisticsfun)
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).
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, , or for the population standard deviation, .
Read Section 11.4 in Chapter 11 of the textbook:
Note:
These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 11.4 of
the textbook.
Introduction to Inference for One Variance (assuming a Normally Distributed Population) (https://www.youtube.com /watch?v=lyd4V8DFCjM&list=UUiHi6xXLzi9FMr9B0zgoHqA&index=2) (jbstatistics)
Inference for a Variance: an example of a Confidence Interval and a Hypothesis Test (https://www.youtube.com
MATH 215 C10 - Study Guide: Unit 5
7
/watch?v=tsLGbpu_NPk&list=UUiHi6xXLzi9FMr9B0zgoHqA) (jbstatistics)
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).
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)
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:
distribution
one-way analysis of variance (ANOVA)
2. use the critical value approach to perform a one-way ANOVA test.
Read the following sections in Chapter 12 of the textbook:
Chapter 12 Introduction
8
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.
These videos provide alternative explanations and further exploration of the concepts and techniques presented in the assigned
textbook readings.
An Introduction to the Distribution (https://www.youtube.com/watch?v=G_RDxAZJ-ug) (jbstatistics)
Using the Table to Find Areas (https://www.youtube.com/watch?v=mSn55vREkIw) (jbstatistics)
Analysis of Variance (ANOVA) Overview in Statistics (https://www.youtube.com/watch?v=CS_BKChyPuc) (mathtutordvd)
ANOVA Basics – The Grand Mean (https://www.youtube.com/watch?v=qSz9xfnXSwg) (mathtutordvd)
How to Calculate and Understand Analysis of Variance (ANOVA) Test (https://www.youtube.com/watch?v=- yQb_ZJnFXw&nohtml5=False) (statisticsfun)
ANOVA: a Visual Introduction (https://www.youtube.com/watch?v=0Vj2V2qRU10) (Brandon Foltz)
One-way ANOVA: a Visual Tutorial (https://www.youtube.com/watch?v=JgMFhKi6f6Y) (Brandon Foltz)
One-way ANOVA: Understanding the Calculation (https://www.youtube.com/watch?v=UrRYITjDOww) (Brandon Foltz)
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.
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
MATH 215 C10 - Study Guide: Unit 5
9
2. The odd-numbered Supplementary Exercises and Advanced Exercises at the end of Chapter 12 (pages 496–498 of the downloadable eText)
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.
The self-test questions are shown here for your information. Download the Unit 5 Self-Test (https://fst-course.athabascau.ca
/science/math/215/r10/self_test/self_test05.html) 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:
a. T F Testing an alternative hypothesis that the mean of the first population is less than the mean of the second population is the same as: .
b. T F 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.
c. T F 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.
d. T F The paired or matched sample test is appropriate to use when the two randomly selected samples are independent.
e. T F In a hypothesis test for independence with a contingency table, the alternative hypothesis is that the two variables are related.
f. T F 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 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 , 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.
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
MATH 215 C10 - Study Guide: Unit 5
10
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 , 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 , can the researcher conclude that the effectiveness of the drug is related to gender? Show all the key steps of the critical value approach.
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 , 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.
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.
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.
MATH 215 C10 - Study Guide: Unit 5
11
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:
a. T F Testing an alternative hypothesis that the mean of the first population is less than the mean of the second population is the same as: .
b. T F 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.
c. T F 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.
d. T F The paired or matched sample test is appropriate to use when the two randomly selected samples are independent.
e. T F In a hypothesis test for independence with a contingency table, the alternative hypothesis is that the two variables are related.
f. T F 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 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 , 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.
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.
MATH 215 C10 - Self-Test: Unit 5
1
At , 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 , can the researcher conclude that the effectiveness of the drug is related to gender? Show all the key steps of the critical value approach.
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 , 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.
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.
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.
MATH 215 C10 - Self-Test: Unit 5
2
Show all your work and keep your calculations to four decimal places, unless otherwise stated.
1. Circle True (T) or False (F) for each of the following:
a. F Testing an alternative hypothesis that the mean of the first population is less than the mean of the second population is the same as: .
b. T 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.
c. F 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.
d. T The paired or matched sample test is appropriate to use when the two randomly selected samples are independent.
e. F In a hypothesis test for independence with a contingency table, the alternative hypothesis is that the two variables are related.
f. F 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 approach and show all key steps.
Solution:
Given and ,
the sample proportion of college graduates unemployed is , and the sample proportion of high-
school graduates unemployed is .
Step 1: State the null hypothesis ( ) and the alternative hypothesis ( )
: ( is the same as )
Step 2: Select the distribution to use.
Select the distribution as the sample sizes are large and the sample sizes multiplied by the sample proportions exceed 5.
Step 3: Calculate the .
T
F
T
F
T
T
-value
-value
1
Step 4: Make a decision.
Since the of 0.0793 exceeds , do not reject .
We cannot conclude that the proportion of college graduates who are unemployed is significantly less than the proportion of high-school graduates who are unemployed.
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 , 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.
Solution:
This is a mean of paired samples problem, where the two samples are dependent.
Let = (cholesterol level after program) – (cholesterol level before program)
To find , set up the following table:
observed
observed
-value
-value
and
2
The hypothesis test:
Step 1: State the null hypothesis ( ) and the alternative hypothesis ( ). Test: (After Before) or (After – Before) , so:
: (cholesterol level did not decrease)
: (cholesterol level decreased)
Since the population is normally distributed with the population standard deviation unknown, the distribution is used.
Step 3: Determine the rejection and non-rejection regions.
Since contains “ ”, this is a one-tailed test with the left tail having an area of .
The .
Step 5: Make a decision.
Since the falls in the rejection region, we reject . Conclusion: The mean cholesterol level has decreased
after taking the exercise program.
observed
observed
3
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 , 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.
Solution:
Step 1: State the null hypothesis ( ) and the alternative hypothesis ( ).
: (mean hospital stay is the same for both populations)
: (mean hospital stay differs between both populations)
Step 2: Select the distribution to use.
Since the populations are normally distributed with population standard deviations unknown, the distribution is used.
Step 3: Determine the rejection and non-rejection regions.
Since contains “ ”, this is a two-tailed test with the combined areas of both rejection regions equal to 0.01. The .
Step 4: Calculate the value of the test statistic.
, where and , where , assuming is true.observed
MATH 215 C10 - Self-Test Answer Key: Unit 5
4
Step 5: Make a decision.
Since falls in the non-rejection region, do not reject . We cannot conclude that there is a
significant difference in the mean hospital stay of insured women versus uninsured women.
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 , can the researcher conclude that the effectiveness of the drug is related to gender? Show all the key steps of the critical value approach.
Solution:
Step 1: State the null hypothesis ( ) and alternative hypothesis ( ).
: The effectiveness of the drug is independent of gender.
: The effectiveness of the drug is related to gender.
Step 2: Select the distribution for tests involving the relation between two categorical variables.
Step 3: Determine the rejection and non-rejection regions.
Since this is a test for independence, it is a right -tailed test with with , where = the number of rows and = the number of columns in
the contingency table given above.
= sample size =
Step 4: Calculate the value of the test statistic.
5
Reject as falls in the rejection region.
Conclusion: The effectiveness of the drug is related to gender.
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 , 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.
Solution:
To test the difference in the mean weight loss between diets, follow the steps below.
total weight loss across all three diets
Between-samples sum of squares
Within-samples sum of squares
:
row total × column total
MATH 215 C10 - Self-Test Answer Key: Unit 5
6
Step 2: Select the distribution to use.
Because we are comparing the means of three normally distributed populations, we use the distribution.
Step 3: Determine the rejection and non-rejection regions.
Since this is a one-way ANOVA test, it is a right-tailed test with with and .
Step 4: Calculate the value of the test statistic.
Step 5: Make a decision.
Since falls in the rejection region, we reject . There is a difference in the mean weight loss between the
three diets.
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.
Solution:
This random sample may be classified as a multinomial experiment. To test whether the observed frequencies for an experiment follow a certain pattern or theoretical distribution, you use a goodness-of-fit test. Use the critical value approach.
To test the hypothesis, follow the steps below.
Step 1: State the null hypothesis ( ) and the alternative hypothesis ( ).
: There is no difference in customer preference between the four types of music.
: There is a difference in customer preference between the four types of music.
numerator denominator
observed MSB
7
Select the distribution for goodness-of-fit tests.
Step 3: Determine the rejection and non-rejection regions.
Since this is a goodness-of-fit test, it is a right-tailed test with with , where = number of categories of music.
Step 4: Calculate the value of the test statistic.
is the probability that a random customer’s preference of music choice is in a given category.
= sample size = customers
determined as follows:
Do not reject as falls in the non-rejection region.
Conclusion: We cannot conclude that there is a difference in customer preference between the four types of music.
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.
Solution:
:
:
Step 2: Select the distribution to use.
Since the population is normally distributed, we use the Chi-square distribution to test a hypothesis about the population variance.
Step 3: Determine the rejection and non-rejection regions.
value
observed
observed
8
© Athabasca University
Since contains “ ”, the rejection region is on the left side of the chi-square distribution, with an area of and .
Step 4: Calculate the value of the test statistic.
Step 5: Make a decision.
Since the test statistic falls in the non-rejection, we do not reject . There is no evidence that the standard deviation of the watch lifetimes is less than months.
observed
9