Mathematics 215: Introduction to Statistics

Study Guide

Unit 4: Estimation and Tests of Hypotheses for One Population

In Unit 3, we discussed probability distributions for both discrete and continuous random variables. At the start of Unit 4, we examine sampling distributions that refer to probability distributions of sample statistics, such as the sample mean and sample proportion. Once you understand the concept of sampling distributions, you will be ready to begin the field of inferential statistics.

The first topic we consider is the Central Limit theorem, which allows us to use the properties of sampling distributions to construct confidence interval estimates and conduct tests of hypotheses involving population means and proportions.

Confidence interval estimation allows us to estimate a population mean or population proportion based on sample data. As an example, the owners of a restaurant could estimate the mean age of all of their customers, based on a sample survey. As a different example, a medical researcher could estimate the proportion of patients who exhibit a specific side-effect when taking a new drug.

Hypothesis testing is used to test a specific claim about a population based on sample data. For example, a sociologist might want to test the claim that, on average, those with master’s degrees make more money than those with bachelor’s degrees. A consumer might want to question a recent advertisement put out by a weight-loss centre that claims to reduce the weight of its clients by at least 10 pounds within a month. A political strategist might want to challenge the view that the political party currently in power will win a majority of the votes in the next election.

Unit 4 of MATH 215 consists of the following sections:

4-1 Mean and Standard Deviation of the Sampling Distribution of the Sample Mean 4-2 Shape of the Sampling Distribution of the Sample Mean 4-3 Mean, Standard Deviation, and Shape of the Sampling Distribution of the Sample Proportion 4-4 Estimation of a Population Mean: Population Standard Deviation Is Known 4-5 Estimation of a Population Mean: Population Standard Deviation Is Unknown 4-6 Estimation of a Population Proportion: Large Samples 4-7 Hypothesis Tests about a Single Population Mean: Population Standard Deviation Is Known 4-8 Hypothesis Tests about a Single Population Mean: Population Standard Deviation Is Unknown 4-9 Hypothesis Tests about a Single Population Proportion: Large Samples

The unit also contains a self-test. When you have completed the material for this unit, including the self-test, complete Assignment 4.

Section 4-1: Mean and Standard Deviation of the Sampling Distribution of the Sample Mean

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:
   • population distribution
   • sampling distribution
   • sampling error and non-sampling error
   • mean and standard deviation of sampling distributions of the sample mean
2. find the mean and standard deviation of the sampling distribution of the sample mean, given the mean and standard deviation of the population distribution, and given the sample size.

Reading

Read the following sections in Chapter 7 of the textbook:

• Chapter 7 Introduction
• Section 7.1
• Section 7.2

Be prepared to read the material in Chapter 7 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 7.1

• Sample vs sampling distributions (Mine Cetinkaya-Rundel)
• Sampling Distributions: Introduction to the Concept (jbstatistics)
• Sampling Distributions (Brian Foltz)
• Variation and Sampling Error (Dr Nic’s Maths and Stats)

Videos Related to Section 7.2

• Dancing Statistics: explaining the statistical concept of ‘sampling’ and ‘standard error’ through dance (BPSOfficial)
• Sampling Distribution of the Sample Mean (jbstatistics)
• Standard Error of the Mean (Brandon Foltz)
  • Note: This is relevant to the situation when a parameter is to be estimated about a population mean from a sample. The standard error is the standard deviation of the sampling distribution of a statistic (for example, a sample mean or a sample proportion). The term may also be used to refer to an estimate of that standard deviation, derived from a particular sample used to compute the estimate.
• Difference Between Standard Deviation and Standard Error (Tommea Analytics)
• Standard Deviation and Standard Error of the Mean (Piers Support)

Exercises

Complete the following exercises from Chapter 7 of the textbook:

• Exercises 7.11, 7.15, and 7.17 on page 283

Remember to show your work as you develop your answers.

Solutions to these exercises are provided in the Student Solutions Manual for Chapter 7 (interactive textbook) and on pages AN10 and AN11 in the Answers to Selected Odd-Numbered Exercises section (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 4-2: Shape of the Sampling Distribution of the Sample Mean

Outcomes

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

1. state the Central Limit theorem and apply it to problems involving sample means.
2. determine the shape of the sampling distribution of the sample mean, given information about the population distribution, the sample size, or both.
3. find the probability that the value of the sample mean will fall within a specified interval, given the population mean, the population standard deviation and the sample size.

Reading

Read the following sections in Chapter 7 of the textbook:

• Section 7.3
• Section 7.4

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 Section 7.3

• Properties of a Sampling Distribution (Steve Mays)

Videos Related to Section 7.3.2:

• An Introduction to the Central Limit Theorem (jbstatistics)
• AP Statistics: Sampling Distributions & the Central Limit Theorem (Math with Mark)
• Sampling Distributions & the Central Limit Theorem (Jennifer Edmonds)

The videos suggested above for Section 7.2 also relate to Section 7.4.

Exercises

Complete the following exercises from Chapter 7 of the textbook (page numbers are for the downloadable eText):

• Exercises 7.23 and 7.25 on page 288
• Exercise 7.27 on page 289
• Exercises 7.33, 7.35, 7.37, 7.39, and 7.41 on page 293

Solutions are provided in the Student Solutions Manual for Chapter 7 (interactive textbook) and on page AN11 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

Section 4-3: Mean, Standard Deviation, and Shape of the Sampling Distribution of the Sample Proportion

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:
   • population proportion and sample proportion
   • sampling distribution of the sample proportion
   • mean and standard deviation of the sampling distribution of the sample proportion
   • Central Limit theorem for sample proportions
2. determine the mean, standard deviation and shape of the sampling distribution of the sample proportion, given the population proportion and the sample size.
3. find the probability that the value of the sample proportion will fall within a specified interval, given the population proportion and the sample size.

Reading

Read the following sections in Chapter 7 of the textbook:

• Section 7.5
• Section 7.6

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 Sections 7.5 and 7.6

• The Sampling Distribution of the Sample Proportion (jbstatistics)
• AP Statistics: Sampling Distributions for Proportions (Michael Porinchak)

Exercises

1. Complete the following exercises from Chapter 7 of the textbook (pages numbers are for the downloadable eText):
   • Exercises 7.55, 7.57, and 7.59 on page 299
   • Exercises 7.63 and 7.65 on pages 301–302

2. Complete the Self-Review Test for Chapter 7 (pages 304–305 of the downloadable eText).

   Solutions are provided in the Student Solutions Manual for Chapter 7 (interactive textbook) and on page AN11 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 7 (pages 303–304 of the downloadable eText)

Section 4-4: Estimation of a Population Mean: Population Standard Deviation Is 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:
   • point estimates and interval estimates
   • significance level
   • confidence level and confidence interval
   • margin of error
2. use the $z$ distribution to construct a confidence interval for the population mean when the population standard deviation is known, the population distribution is normal and the sample size is small ( $<30$ ).
3. use the $z$ distribution to construct a confidence interval for the population mean when the population standard deviation is known and the sample size is large ( $\ge 30$ ).
4. compute the sample size that will be required to estimate the mean, given the confidence level, the population standard deviation and a specified margin of error.

Reading

Read the following sections in Chapter 8 of the textbook:

• Chapter 8 Introduction
• Section 8.1
• Section 8.2

Be prepared to read the material in Chapter 8 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 Chapter 8

• Sample Mean Proximity to Population Mean (Brandon Foltz)
• Understanding Confidence Intervals (Dr Nic’s Maths and Stats)

Videos Related to Section 8.1

• Point Estimators (Brandon Foltz)
• Introduction to Confidence Intervals (Kenneth Strazzeri)
• Margin of Error and Confidence Intervals (Kenneth Strazzeri)

Videos Related to Section 8.2

• Confidence Interval Assumptions (Kenneth Strazzeri)
• $z$ Scores – Statistics (Math Meeting) [for when $\sigma$ is known]
• Confidence Intervals for a Population Mean ($\sigma$ known) (Joshua Emmanuel)
• Confidence Interval Estimation, Sigma Known (Brandon Foltz)
• Intro to Confidence Intervals for One Mean (Sigma Known) (jbstatistics)

Exercises

Complete the following exercises from Chapter 8 of the textbook (page numbers are for the downloadable eText):

• Exercises 8.11, 8.13, 8.15, 8.19, 8.23, and 8.25 on pages 322–323

  Solutions are provided in the Student Solutions Manual for Chapter 8 (interactive textbook) and on page AN12 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

Section 4-5: Estimation of a Population Mean: Population Standard Deviation Is Unknown

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:
   • $t$ distribution
   • sample standard deviation
2. use the $t$ distribution to construct a confidence interval for the population mean when the population standard deviation is unknown, the population distribution is normal and the sample size is small ( $<30$).
3. use the $t$ distribution to construct a confidence interval for the population mean when the population standard deviation is unknown and the sample size is large ( $\ge 30$).

Reading

Read Section 8.3 in Chapter 8 of the downloadable eText.

Supplementary Video Resources

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

Videos Related to Section 8.3

• $t$ Scores – Statistics (Math Meeting) [for when $\sigma$ is unknown]
• Confidence Interval Assumptions (Kenneth Strazzeri)
• Confidence Intervals from Repeated Samples (sigma unknown) (Kenneth Strazzeri)
• Confidence Intervals for One Mean: Sigma Not Known ($t$ Method) (jbstatistics)
• Confidence Interval Concepts, Sigma Unknown (Brandon Foltz)
• Confidence Interval Problems, Sigma Unknown (Brandon Foltz)

Exercises

Complete the following exercises from Chapter 8 of the textbook (page numbers are for the downloadable eText):

• Exercises 8.33, 8.35, and 8.37 on page 329
• Exercises 8.41, 8.43, and 8.45 on page 330

Solutions are provided in the Student Solutions Manual for Chapter 8 (interactive textbook) and on page AN12 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

Section 4-6: Estimation of a Population Proportion: Large Samples

Outcomes

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

1. define and apply the “estimator of the standard deviation of the sampling distribution of the sample proportion.”
2. use the $z$ distribution to construct a confidence interval for the population proportion, given sample data.
3. compute the sample size that will be required to estimate the proportion, given the level of confidence and a specified margin of error.

Reading

Read Section 8.4 in Chapter 8 of the textbook.

Supplementary Video Resources

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

Videos Related to Section 8.4

• Confidence Intervals for Categorical Data (Kenneth Strazzeri)
• Confidence Intervals for a Proportion: Determining the Minimum Sample Size (jbstatistics)

Exercises

1. Complete the following exercises from Chapter 8 of the textbook (page numbers are for the downloadable eText):
   • Exercise 8.53 on page 335
   • Exercises 8.57, 8.59, 8.61, and 8.63 on page 336
   • Exercises 8.67 and 8.69 on page 337
   • Supplementary Exercises 8.77, 8.79, 8.81, 8.83, and 8.85 on pages 338–339

2. Complete the Self-Review Test for Chapter 8 (pages 339–340 of the downloadable eText). Omit questions 14 and 15.

   Solutions are provided in the Student Solutions Manual for Chapter 8 (interactive textbook) and on page AN12 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 8 (page 339 in the eText)

Section 4-7: Hypothesis Tests about a Single Population Mean: Population Standard Deviation Is 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:
   • null hypothesis
   • alternative hypothesis
   • critical value
   • Type I error
   • level of significance
   • Type II error
   • two-tailed test
   • left-tailed test
   • right-tailed test
   • test statistic or observed value
   • statistically significantly different and statistically not significantly different
   • $p\text{-value}$
2. use the critical value approach to perform a hypothesis test about the population mean, given the population standard deviation and sample data.
3. use the $p\text{-value}$ approach to perform a hypothesis test about the population mean, given the population standard deviation and sample data.

Reading

1. Read the following sections in Chapter 9 of the textbook:
   • Chapter 9 Introduction
   • Sections 9.1
   • Section 9.2

2. Read Additional Topics 4A, 4B, and 4C in this Study Guide, below.

   Important: Complete this reading before you complete the exercises for this section.

Be prepared to read the material in Chapter 9 and the additional topics 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 Chapter 9

• Intro to Hypothesis Testing - an Overview (Kevin Martz)
• Choosing Which Statistical Test to Use (Dr Nic’s Maths and Stats)
• Relationship Between Hypothesis Tests and Confidence Intervals (two-sided) (jbstatistics)

Videos Related to Section 9.1

• Intro to Hypothesis Testing in Statistics – Problems & Examples (Math and Science)
• An Introduction to Hypothesis Testing (jbstatistics)
• One-Tailed and Two-Tailed Hypotheses (Roger Morrissette)
• One-Sided Test or Two-Sided Test? (jbstatistics)
• Understanding the $p\text{-value}$ (Dr Nic’s Maths and Stats)
• What is a $p\text{-value}$? (jbstatistics)
• Important Statistical Concepts: significance, strength, association, causation (Dr Nic’s Maths and Stats)
• Visualizing Type I and Type II Errors (Brandon Foltz)
• Type I and Type II Errors (Brandon Foltz)
• Type I Errors, Type II Errors and the Power of the Test (jbstatistics)
• What Factors Affect the Power of a $z$ test? (jbstatistics)
• Calculating the Power and the Probability of a Type II Error (one-tailed $z$-test example) (jbstatistics)
• Calculating the Power and the Probability of a Type II Error (two-tailed $z$-test example) (jbstatistics)
• To $z$ or to $t$ , That is the Question (Brandon Foltz)
• $z\text{-test}$ versus $t\text{-test}$ (Math Meeting)
• Hypothesis Tests on One Mean: $t\text{-Test}$ or $z\text{-Test}$? (jbstatistics)

Videos Related to Section 9.2

• $Z\text{-Tests}$ for One Mean: Introduction (jbstatistics) [for when $\sigma$ is known]
• Single Sample Hypothesis $Z\text{-Test}$ Concepts (Brandon Foltz)
• Single Sample Hypothesis $Z\text{-Test}$ Examples (Brandon Foltz)
• Single Sample Hypothesis $Z\text{-Test}$ Alpha and $p\text{-values}$ (Brandon Foltz)
  • This video gives a relative comparison of differing alpha values and their corresponding critical $z\text{-values}$ and $p\text{-values}$ of the test statistic.

Videos Related to Section 9.2.1

• Calculate the $p\text{-Value}$ in Statistics – Formula to Find the $p\text{-Value}$ in Hypothesis Testing (Math and Science)
  • One helpful comment from this video is “If your $p\text{-value}$ is low, reject the $H_0$.”
• What is a $p\text{-value}$? (jbstatistics)
• $z\text{-Tests}$ for One Mean: the $p\text{-value}$ (jbstatistics)
• $z\text{-Tests}$ for One Mean: an example (jbstatistics)
• Hypothesis Testing ($p\text{-value}$ Method) (poysermath)
• Hypothesis Testing using the $p\text{-value}$ Method (Example) (poysermath)
• A Tool for Calculating the $p\text{-value}$ for Various Hypothesis Tests (statdistributions.com)
• Statistical Significance versus Practical Significance (jbstatistics)

Videos Related to Section 9.2.2

• $z\text{-Tests}$ for One Mean: The Rejection Region Approach (jbstatistics)
  • Note: The rejection region approach is another name for the critical value approach
• Determining Critical Values and Rejection Regions (Clutch Prep)
• Stats: Hypothesis Testing (Traditional Method) (poysermath)

Exercises

Once you have completed all the reading, including Additional Topics 4A, 4B, and 4C, complete the following exercises from Chapter 9 of the textbook (page numbers are for the downloadable eText):

• Exercises 9.5 and 9.7 on page 354
• Exercises 9.15, 9.17, and 9.21 on page 365
• Exercises 9.25, 9.27, 9.29, and 9.31 on page 366

Solutions are provided in the Student Solutions Manual for Chapter 9 (interactive textbook) and on pages AN12 and AN13 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

Required Reading: Additional Topic 4A: The $\mathit{p}\mathbf{\text{-Value}}$ Approach

Key Steps in the $p\text{-Value}$ Approach

Unless otherwise stated in the exercise/problem you are working on, make sure that you show your work regarding all four steps in the $p\text{-value}$ approach, as follows:

Step 1: State the null hypothesis ($H_0$) and the alternative hypothesis ( $H_1$).
Step 2: Select the distribution to use.
Step 3: Calculate the $p\text{-value}$.
Step 4: Make a decision.

The $\mathit{p}\mathbf{\text{-value}}$ (or probability value) is the probability of getting a sample statistic (such as the sample mean or its related $z$ value) or a more extreme sample statistic in the direction of the alternative hypothesis when the null hypothesis is true.

For a one-tailed test, the $p\text{-value}$ is given by the area in the tail of the sampling distribution curve beyond the observed value of the sample statistic (or its related $z$ value).

The figure below, reproduced from your text, shows the $p\text{-value}$ for a right-tailed test about $\mu$, where $H_1$ has a “$>$” sign.

Figure 9.5: $p\text{-Value}$ for a Right-Tailed Test
Source: Prem S. Mann, Introductory Statistics, 9th ed. (Wiley, 2016) [VitalSource], 356. This material is reproduced with the permission of John Wiley & Sons Canada, Ltd.

Mann explains that “for a left-tailed test, the $p\text{-value}$ will be the area in the lower tail of the sampling distribution curve to the left of the observed value” of the sample mean (or $z$) (Mann 356).

Components of Step 4 (“Make a decision”)

Step 4 comprises two components. You must complete both of these components in order to complete Step 4 and state your decision properly.

Note: you will not receive any marks for completing Step 4 of a hypothesis test unless you complete both of these components.

Step 4, Component 1

For the given problem/exercise, display a comparison of the computed $p\text{-value}$ from Step 3 with the given level of significance. Based on this comparison, state “reject the null hypothesis” or “do not reject the null hypothesis” by applying the following rule:

If the $p\text{-value}\le \alpha$, reject $H_0$.

If the $p\text{-value}>\alpha$, do not reject $H_0$.

To further explain the rule above, if the $p\text{-value}$ is relatively low, this means that the probability of generating the sample mean observed in the problem is low, assuming that $H_0$ is true. More likely, $H_0$ is not true, so the null hypothesis should be rejected.

Step 4, Component 2

Based on your decision to reject or not reject the null hypothesis, state the conclusion in terms of the practical context of the problem/exercise at hand. For example, if the test of hypothesis relates to average income, your stated conclusion should be in terms of average income; or, if the test of hypothesis relates to mean weight, your stated conclusion should be in terms of mean weight.

Required Reading: Additional Topic 4B: The Critical-Value Approach

Key Steps in the Critical Value Approach

Unless otherwise stated in the exercise/problem you are working on, make sure that you show your work regarding all five steps in the critical-value approach, as follows:

Step 1: State the null hypothesis ( $H_0$) and the alternative hypothesis ( $H_1$).
Step 2: Select the distribution to use.
Step 3: Determine the rejection and non-rejection regions (critical values, etc.).
Step 4: Calculate the value of the test statistic.
Step 5: Make a decision.

Graph Related to Step 3

We strongly encourage you to sketch the appropriate graph illustrating the rejection and non-rejection regions, as this will help you to correctly determine the critical values.

Components of Step 5 (“Make a decision”)

Step 5 comprises two components. You must complete both of these components in order to complete Step 5 and state your decision properly.

Note: you will not receive any marks for completing Step 5 of a hypothesis test unless you complete both of these components.

Step 5, Component 1

For the given problem/exercise, display a comparison of the computed test statistic from Step 4 with the determined rejection/non-rejection regions in Step 3. Based on this comparison, state “reject the null hypothesis” or “do not reject the null hypothesis” by applying the following rule:

If the test statistic falls inside the rejection region, reject $H_0$.

If the test statistic falls outside the rejection region, do not reject $H_0$.

Step 5, Component 2

Based on your decision to reject or not reject the null hypothesis, state the conclusion in terms of the practical context of the problem/exercise at hand. For example, if the test of hypothesis relates to average income, your stated conclusion should be in terms of average income; or, if the test of hypothesis relates to mean weight, your stated conclusion should be in terms of mean weight.

Required Reading: Additional Topic 4C: The $\mathit{p}\mathbf{\text{-Value}}$ and Critical Value Approaches

There is a one-to-one correspondence between the $p\text{-value}$ approach to hypothesis testing and the critical value approach:

• if the $p\text{-value}$ for a test of hypothesis is less than α , then the observed value of the test statistic will fall in the rejection region of the critical value approach, and consequently $H_0$ will be rejected;
• if the observed value of the test statistic falls in the rejection region of the critical value approach, then the $p\text{-value}$ will be less than α , and again $H_0$ will be rejected.

Most statistical software packages perform tests of hypotheses using a $p\text{-value}$ approach rather than a critical value approach. Our experience has shown, however, that students find the critical value approach more “user-friendly” (i.e., understandable) than the $p\text{-value}$ approach.

An advantage of using the $p\text{-value}$ approach rather than the critical value approach is that with this approach you are able not only to decide whether to reject or not reject $H_0$, but also to get a sense of how significant the decision/conclusion is (that is, how strong the evidence is to support the decision to reject or not reject $H_0$). This is further explained below.

The following table provides guidelines to interpreting $p\text{-values}$ when you encounter them in future research.

In essence, a null hypothesis ( $H_0$ ) is a claim that is “on trial.” It represents the status quo in a given situation, which is considered innocent until proven guilty beyond a reasonable doubt. In medical research, an $H_0$ may be that a drug or treatment has “no” effect; in business research, an $H_0$ may be that an advertising program has “no” effect. As the table above shows, very small $p\text{-values}$ provide strong evidence that a drug or treatment does have an effect, or that an advertising program is indeed effective after all.

Section 4-8: Hypothesis Tests about a Single Population Mean: Population Standard Deviation Unknown

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 about the population mean, given sample data, when the population standard deviation is unknown.

Reading

1. Read Section 9.3 in Chapter 9 of the textbook.

2. Read Additional Topics 4D and 4E in this Study Guide, below.

   Important: Complete this reading before you complete the exercises for this section.

Supplementary Video Resources

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

Videos Related to Section 9.3

• Introduction to the $t$ Distribution (non-technical) (jbstatistics)
• An Introduction to the $t$ Distribution (mathematical) (jbstatitics)
• $t\text{-Tests}$ for One Mean: Introduction (jbstatistics)
• $t\text{-Tests}$ for One Mean: investigating the normality assumption (jbstatistics)
• Single Sample Hypothesis $t\text{-test}$ Concepts (Brandon Foltz)
• Single Sample Hypothesis $t\text{-test}$ Examples (Brandon Foltz)
• $t\text{-Tests}$ for One Mean: an example (jbstatistics)
  • Note: The content of this video is relevant only up to the 7:35 minute mark.

Exercises

Complete the following exercises from Chapter 9 of the textbook (page numbers are for the downloadable eText):

• Exercises 9.35, 9.37, 9.41, 9.43, 9.45, and 9.49 on pages 374–375

  Solutions are provided in the Student Solutions Manual for Chapter 9 (interactive textbook) and on page AN13 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

Required Reading: Additional Topic 4D: Estimating the $\mathit{p}\mathbf{\text{-Value}}$ for the $\mathit{t}$ Distribution of Two-Tailed Tests

This reading takes Example 9-5 from the textbook, which you have already read, and adds a more detailed explanation of estimating the $p\text{-value}$ for a two-tailed $t$ test, as opposed to a similar test involving the $z$ distribution.

To test the hypothesis and to make the decision, we apply the following four steps:

The $t$ value (also called the test statistic) is:

${\displaystyle t=\frac{(\overline{x}-\mu )}{s_{\overline{x}}}=\frac{\left(12.9–12.5\right)}{0.1886}=2.121}$, so $\pm 2.121$ (two-tailed)

The $p\text{-value}$ is the area under the $t$ distribution curve beyond “$t$, ” which is $\pm 2.121$, as shown below:

Figure 9.11: The Required $p\text{-Value}$
Source: Prem S. Mann, Introductory Statistics, 9th ed. (Wiley, 2016) [VitalSource], 369. This material is reproduced with the permission of John Wiley & Sons Canada, Ltd.

In determining the $p\text{-value}$ related to $t$, the best thing to do is to use Table V in Appendix B of the textbook to find the range that contains the $p\text{-value}$ (i.e., estimate the $p\text{-value}$), as explained below.

Table V The $\mathit{t}$ Distribution Table

The entries in this table give the critical values
of $t$ for the specified number of degrees
of freedom and areas in the right tail.

Table V: The $t$ Distribution Table (Excerpt)
Source: Adapted from Prem S. Mann, Introductory Statistics, 9th ed. (Wiley, 2016) [VitalSource], B21. This material is reproduced with the permission of John Wiley & Sons Canada, Ltd.

Steps for Estimating the $p\text{-Value}$

1. Read down the $t$ Distribution Table (above) until you find the appropriate degrees of freedom, which in this case are: $df=n-1=18-1=17$.
2. Locate the calculated $t$ value of 2.121 in the row with 17 degrees of freedom. It falls between 2.110 and 2.567.
3. Read to the top of the table to locate the area to the right of this calculated $t$ value. The area to the right is between 0.025 and 0.01. This range of area is one-half the desired $p\text{-value}$, because this is a two-tailed hypothesis test.

4. Since we have a two-tailed test, multiply this range of areas by two to get the range of the desired $p\text{-value}$, as follows:

   Estimated $p\text{-value}$: $p\text{-value}$ is between 2(0.01) and 2(0.025)
   Estimated $p\text{-value}$: $p\text{-value}$ is between 0.02 and 0.05

Step 4. Make a decision.

Since the estimated $p\text{-value}$ exceeds $\text{alpha}=0.01$, we do not reject $H_0$. Therefore, we cannot conclude that the mean walking age is different from 12.5 months.

Required Reading: Additional Topic 4E: Estimating the $\mathit{p}\mathbf{\text{-Value}}$ for the $\mathit{t}$ Distribution of One-Tailed Tests.

This reading takes Example 9-6 from the textbook, which you have already read, and adds a more detailed explanation of estimating the $p\text{-value}$ for a one-tailed $t$ test, as opposed to a similar test involving the $z$ distribution.

The $t$ value (that is, the test statistic, or simply $t$) is:

${\displaystyle s_{\overline{x}}=\frac{s}{\sqrt{n}}=\frac{3}{\sqrt{45}}=.44721360}$

${\displaystyle t=\frac{\overline{x}-\mu }{s_{\overline{x}}}=\frac{63.4-65}{.44721360}=-3.578}$From $H_0$

$df=n-1=45-1=44$

The $p\text{-value}$ is the area under the $t$ distribution curve beyond $t=-3.578$, as shown below:

Figure 9.12: The Required $p\text{-Value}$
Source: Prem S. Mann, Introductory Statistics, 9th ed. (Wiley, 2016) [VitalSource], 370. This material is reproduced with the permission of John Wiley & Sons Canada, Ltd.

In determining the $p\text{-value}$ related to $t$, the best thing to do is to use Table V in Appendix B in the textbook to find the range that contains the $p\text{-value}$ (i.e., estimate the $p\text{-value}$) as explained below.

Table V The $t$ Distribution Table (continued)

The entries in this table give the critical values
of $t$ for the specified number of degrees
of freedom and areas in the right tail.

Table V: The $t$ Distribution Table (Excerpt)
Source: Adapted from Prem S. Mann, Introductory Statistics, 9th ed. (Wiley, 2016) [VitalSource], B21–B22. This material is reproduced with the permission of John Wiley & Sons Canada, Ltd.

Steps in Estimating $p\text{-Value}$

1. Read down the $t$ Distribution Table (above) until you find the appropriate degrees offreedom, which in this case are: $df=n-1=44$.
2. Ignoring the sign of the calculated test statistic, locate it in the row with 44 degrees of freedom. It falls to the right of 3.286.
3. Read to the top of the table to locate the area to the right of this calculated $t$ value. The area to the right is less than 0.001. Since the $t$ distribution is symmetric, the area to the left of $t$ is also less than 0.001.
4. Since we have a one-tailed test, this estimated area to the left of $t=-3.578$ is our estimated $p\text{-value}$. That is, $p\text{-value}<0.001$ .

Step 4. Make a decision.

Since the estimated $p\text{-value}$ of 0.001 is less than $\text{alpha}=0.025$ , we reject $H_0$. Therefore, we can conclude that the mean life of such batteries is less than 65 months.

Section 4-9: Hypothesis Tests about a Single Population Proportion: Large Samples

Outcomes

After completing the readings and exercises for this section, you should be able to do the following: Use the critical value approach and the $p\text{-value}$ approach to perform a hypothesis test about the population proportion, given data from a large sample.

Reading

Read Section 9.4 in Chapter 9 of the textbook.

Supplementary Video Resources

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

Videos Related to Section 9.4

• An Introduction to Inference for a Proportion (jbstatistics)
• Inference for One Proportion: an example for a Confidence Interval and a Hypothesis Test (jbstatistics)
• Inference for a Population Proportion (Bryan Nelson)

Exercises

1. Complete the following exercises from Chapter 9 of the textbook (page numbers are for the downloadable eText):
   • Exercises 9.53, 9.55, 9.57, 9.61, 9.63, and 9.65 on page 383
   • Supplementary Exercises 9.73, 9.75, 9.79, 9.81, and 9.83 on pages 386–387
     • Note: For Exercises 9.75, 9.79, and 9.83, use the critical value approach.

2. Complete the problems in the Self-Review Test for Chapter 9 (pages 388–389 of the downloadable eText). If a problem asks you to conduct a test of hypothesis and does not specify which approach to use, use the critical value approach.

   Solutions are provided in the Student Solutions Manual for Chapter 9 (interactive textbook) and on pages AN13 and AN14 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

3. Complete the Unit 4 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 and Supplementary Exercises that are not assigned above
2. The odd-numbered Advanced Exercises at the end of Chapter 9 (page 387 in the eText)

Assignment 4

Once you have completed the Unit 4 Self-Test below, complete Assignment 4. 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 4.

Unit 4 Self-Test

The self-test questions are shown here for your information. Download the Unit 4 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

      The standard deviation of the sampling distribution of the sample mean is equal to the population standard deviation.
   2. TF

      If the population distribution is positively skewed, then the sampling distribution of the sample mean is also positively skewed.
   3. TF

      When the population standard deviation is unknown and the sample size exceeds 30, the $z$ distribution is used to compute a confidence interval for the population mean.
   4. TF

      When the population standard deviation is known and the sample size exceeds 30, the $z$ distribution is used to compute a confidence interval for the population mean.
   5. TF

      A larger sample size will tend  to reduce the width of a confidence interval.
   6. TF

      In conducting a test of hypothesis, if the $p\text{-value}$ exceeds the level of significance, we reject the null hypothesis.
   7. TF

      In conducting a test of hypothesis, if the alternative hypothesis consists of a “$<$” expression, the critical value will be a negative number.
   8. TF

      In conducting a test of hypothesis, if the $p\text{-value}$ is less than 0.001, the evidence against the null hypothesis is considered to be very weak.

2. Past census surveys in a large Canadian province indicate that 40% of provincial voters favor the implementation of a carbon tax to combat global warming.

   Consider a sampling (random) experiment where 100 voters are selected at random and the sample proportion of voters who favor a carbon tax is to be observed.

   1. What would be the shape of the sampling distribution of the sample proportion in favor of the carbon tax, and why?
   2. Determine the mean of the sampling distribution of the sample proportion in favor of the carbon tax.
   3. Determine the standard deviation of the sampling distribution of the sample proportion in favor of the carbon tax.
   4. Find the probability (to 4 decimal places) that, in the random sample of 100 voters, the sample proportion who favor a carbon tax is:
      1. less than 0.30
      2. between 0.30 and 0.40
   5. If, among the 100 voters selected at random, 44 are in favor of a carbon tax, compute the sampling error. Assume that there are no non-sampling errors.

3. Recent studies involving all students in a community college found that these students spend an average of 20 hours a week on homework outside of the classroom, with a standard deviation of 4 hours per week. Assume that the data collected follows a normal distribution.

   If a random sample of 25 students from this community college is selected, find the probability (to 4 decimals) that the sample mean weekly homework hours will be

   1. at least 22 hours.
   2. between 18 and 22 hours.
   3. less than 10 hours per week.
4. What is the minimum sample size needed for a 99% confidence interval estimate for the population proportion to have a maximum margin of error of 0.06
   1. if there is a preliminary estimate of 0.80?
   2. if there is no preliminary estimate, so the most conservative estimate must be used?

5. In a recent municipal survey, 2,000 randomly selected taxpayers were sampled and 1,200 adults stated that they are in favor of constructing a new hockey arena.

   Construct a 90% confidence interval (calculated to 4 decimal places) to estimate the percentage of all municipal taxpayers that are in favor of constructing the hockey arena.

6. Past market research indicates that the ages of all the regular customers of a large fitness club are normally distributed. A recent sample of 6 randomly selected regular customers resulted in the following stem-and-leaf display of the ages of the selected customers:

   1	8	9
   2	2	4	6
   3	3

   Construct a 95% confidence interval estimate for the population mean age of all the club’s regular customers.

7. A medical researcher wishes to estimate, within 2 points, the average systolic blood pressure of university students located in a Canadian province. If the researcher wishes to be 96% confident, how large a sample should she select if the population standard deviation systolic blood pressure for all the provincial university students is 6.0?

8. A census survey indicates that the national average family size was 3.25 persons per family in 2015. A 2018 sample of families randomly selected across the country results in the following family sizes:
   4, 2, 3, 2, 1, 3, 4, 2, 5, 4

   Assuming that the population of family sizes is normally distributed, conduct a test of hypothesis at the 5% level to determine if the average family size has decreased between 2015 and 2018.

   1. Show all key steps using the $p\text{-value}$ approach.
   2. Show all key steps using the critical value approach.
9. A large online retail company claims that more than 80% of all its orders are delivered to customers’ homes within 72 hours. A researcher working for the Department of Consumer and Corporate Affairs, suspicious of this claim, took a random sample of 400 orders and found that 330 of them were delivered to homes within a 72 hour period. Conduct a test of hypothesis at the 1% level to determine if the random sample supports the retailer’s claim.
   1. Show all key steps using the $p\text{-value}$ approach.
   2. Show all key steps using the critical value approach.
10. In 2014 the average cost of all weddings in the country was $23,000. A recent sample of 64 couples who got married this year produced a mean wedding cost of $24,500 with a standard deviation of $4,400. Conduct a test of hypothesis at the 5% level to determine if the average cost of weddings has changed.
    1. Show all key steps using the $p\text{-value}$ approach.
    2. How strong is the evidence against the null hypothesis ( $H_0$)? Explain your reasoning. (See Additional Topic 4C: The $p\text{-Value}$ and Critical Value Approaches in Unit 4 of the Study Guide, Section 4-7.)

References

Mann, Prem S. Introductory Statistics, 9th ed. Wiley, 2016. [VitalSource].