Mathematics 216 Computer-oriented Approach to Statistics

Study Guide :: Unit 4

Inference on One Sample

Introduction

In Unit 3, you learned about the probability distribution of a random variable and how to compute the associated mean, variance, and probabilities. Numerical quantities that describe probability distributions are called parameters. In practice, because this information is not available it must be estimated using statistical techniques.

The most accurate way of obtaining information about a population parameter would be to collect the relevant data from every member of that population. Such a procedure is impractical in most cases. For example, a tire manufacturer who wanted to know the average lifespan of their tires could not stay in business if they tested every tire until it wore out. Similarly, if we wanted to know the average lifespan of Canadians, we could not wait until all members of the Canadian population had died.

Hence, the best way to gather information about a population is to collect data from a representative sample of the population and make inferences about the population. The numerical descriptive measures (such as sample mean, sample proportion, sample standard deviation, etc.) are called statistics. However, statistics vary from sample to sample. Stated simply, if we consider all possible samples from a given population, we will find variability in the sample statistic; that is, each sample statistic will have its own distribution. If all possible values of a sample statistic that might occur are organized into a probability distribution, the resulting distribution is called the sampling distribution.

This unit begins by discussing sampling distribution of means and proportions, and how the mean and standard deviation are related to the mean and standard deviation of the parent population. Central limit theorem, the foundation for the inferential branch of statistics, is introduced in this context. Once you understand the concept of sampling distributions and central limit theorem, you are ready to begin the study of inferential statistics, which is concerned with estimating a population parameter (characteristic) based on results observed from a sample.

In the rest of this unit, we discuss the two categories of inferential statistics: estimation and hypothesis testing. Estimation is the process of obtaining a single numerical value (point estimate) or a set of values (interval estimate or confidence interval) intended as a “best guess” of the unknown population parameter. In hypothesis testing, we test claims regarding a characteristic of one or more populations. The claims that we test concern the population mean and the population proportion.

Sampling Distributions and the Central Limit Theorem

Learning Objectives

After completing the readings and exercises assigned for this topic, you should be able to:

  1. Explain the term “sampling distribution” and verify its properties.
  2. Calculate the mean and standard deviation of the sampling distribution of sample means.
  3. Describe and interpret central limit theorem.
  4. Finding Probabilities for the Sample Mean.

Important Note: For help accessing the eText resources referred to below, see Navigating Your eText on the course home page.

Required Reading

Elementary Statistics, Chapter 5, Section 5.4 Sampling Distributions and the Central Limit Theorem

Try It Yourself Examples

Work through each Try It Yourself example in this section of the eText. Check your work against the solutions provided.

Exercises in Your eText

Do the following exercises in your eText:

Chapter 5, Section 5.4 Exercises 1, 5, 7, 9, 11, 13, 15. Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 5 Section 5.4 are available on the textbook publisher’s MyLab website.

Chapter 5.4 Review (Extra Online Practice)

For more practice working with the topics in this chapter of the eText, work through this review. Or, if you feel you have mastered this material, you may skip to the computer lab section of this unit.

Review Learning Objectives

Before proceeding to the online exercises, briefly review the Learning Objectives for the topic (below) presented in the previous section of this Study Guide:

  • Sampling Distributions and the Central Limit Theorem
Optional Practice in the MyLab Study Plan

For more practice on the topics/sections of this chapter of your eText, visit Pearson MyLab, and work interactively through the exercises in the Study Plan. For help accessing this resource, see Accessing Pearson MyLab on the course home page.

Confidence Intervals for the Mean (σ Known)

Learning Objectives

After completing the readings and exercises assigned for this topic, you should be able to:

  1. Explain the meaning of the key terms:
    • point estimate; interval estimate
    • confidence interval; level of confidence
    • margin of error
  2. Compute point estimate and margin of error for the population mean.
  3. Construct and interpret intervals for the population mean.
  4. Determine the minimum sample size required when estimating μ.

Important Note: For help accessing the eText resources referred to below, see Navigating Your eText on the course home page.

Required Reading

Elementary Statistics, Chapter 6, Section 6.1 Confidence Intervals for the Mean (σ Known)

Try It Yourself Examples

Work through each Try It Yourself example in this section of the eText. Check your work against the solutions provided.

Exercises in Your eText

Do the following exercises in your eText:

Chapter 6, Section 6.1 Exercises 3, 35, 37, 41, 49, 55. Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 6, Section 6.1 are available on the textbook publisher’s MyLab website.

Confidence Intervals for the Mean (σ Unknown)

Learning Objectives

After completing the readings and exercises assigned for this topic, you should be able to:

  1. Interpret t distribution and use a t-distribution table.
  2. Know the properties of students’ t-distribution.
  3. Construct confidence intervals when n < 30, the population is normally distributed, and σ is known.

Important Note: For help accessing the eText resources referred to below, see Navigating Your eText on the course home page.

Required Reading

Elementary Statistics, Chapter 6, Section 6.2 Confidence Intervals for the Mean (σ Unknown)

Try It Yourself Examples

Work through each Try It Yourself example in this section of the eText. Check your work against the solutions provided.

Exercises in Your eText

Do the following exercises in your eText:

Chapter 6, Section 6.2 Exercises 1, 3, 5, 7, 9, 13, 17, 21. Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 6 Section 6.2 are available on the textbook publisher’s MyLab website.

Confidence Intervals for Population Proportions

Learning Objectives

After completing the readings and exercises assigned for this topic, you should be able to:

  1. Obtain a point estimate for the population proportion.
  2. Construct and interpret confidence intervals for the population proportion.
  3. Determine the minimum sample size required for estimating a population proportion within a specified margin of error.

Important Note: For help accessing the eText resources referred to below, see Navigating Your eText on the course home page.

Required Reading

Elementary Statistics, Chapter 6, Section 6.3 Confidence Intervals for Population Proportions

Try It Yourself Examples

Work through each Try It Yourself example in this section of the eText. Check your work against the solutions provided.

Exercises in Your eText

Do the following exercises in your eText:

Chapter 6, Section 6.3 Exercises 1, 3, 7, 13, 19, 21, 23, 25. Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 6 Section 6.3 are available on the textbook publisher’s MyLab website.

Chapter 6 Review (Extra Online Practice)

For more practice working with the topics in this chapter of the eText, work through this review. Or, if you feel you have mastered this material, you may skip to Computer Lab 4A.

Review Learning Objectives

Before proceeding to the online exercises, briefly review the Learning Objectives for each of the following topics (listed below), which are presented in previous sections of this Study Guide.

  • Confidence Interval for Mean (σ Known)
  • Confidence Interval for Mean (σ Unknown)
  • Confidence Interval for Population Proportion
Optional Practice in the MyLab Study Plan

For more practice on the topics/sections of Chapter 6, visit Pearson MyLab, and work interactively through the exercises in the Study Plan. For help accessing this resource, see Accessing Pearson MyLab on the course home page.

Computer Lab 4A

In Computer Lab 4A, you will learn to use StatCrunch to develop solutions to exercises related to topics in the eText’s Chapter 5 and 6.

Computer Lab 4A Detailed Instructions

Your Computer Lab activities and the detailed step-by-step instructions (Guided Solutions) that will guide you in using StatCrunch to complete these are in the Computer Lab 4A file.

Computer Lab 4A Quick Reviews

The Quick Reviews (QRs) summarize a few key steps (but not all steps) needed to complete each Activity in Computer Lab 4A. These QRs will be useful when you are preparing for the computer components of the assignments, midterm exam, and final exam. To access the QRs, click Computer Lab 4A QRs.

Introduction to Hypothesis Testing with One Sample

Learning Objectives

After completing the readings and exercises assigned for this topic, you should be able to:

  1. Determine the null and alternative hypotheses from a claim.
  2. Distinguish between type I and type II errors.
  3. Interpret the level of significance.
  4. Determine whether to use a one-tailed or two-tailed statistical test.
  5. Compute and interpret P-value.
  6. Make and interpret a decision based on the results of a hypothesis test.

Important Note: For help accessing the eText resources referred to below, see Navigating Your eText on the course home page.

Required Reading

Elementary Statistics, Chapter 7, Section 7.1 Introduction to Hypothesis Testing

Try It Yourself Examples

Work through each Try It Yourself example in this section of the eText. Check your work against the solutions provided.

Exercises in Your eText

Do the following exercises in your eText:

Chapter 7, Section 7.1 Exercises 1, 11, 13, 15,  21, 29, 31, 33, 35, 37, 41, 43, 45, 51. Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 7 Section 7.1 are available on the Pearson MyLab website.

Hypothesis Testing for the Mean (σ Known)

Learning Objectives

After completing the readings and exercises assigned for this topic, you should be able to:

  1. Use P-values to make decisions.
  2. Use P-values in a z-test.
  3. Construct critical (rejection) regions and critical values.
  4. Use rejection regions in a z-test.

Important Note: For help accessing the eText resources referred to below, see Navigating Your eText on the course home page.

Required Reading

Elementary Statistics, Chapter 7, Section 7.2 Hypothesis Testing for the Mean (σ Known)

Try It Yourself Examples

Work through each Try It Yourself example in this section of the eText. Check your work against the solutions provided.

Exercises in Your eText

Do the following exercises in your eText:

Chapter 7, Section 7.2 Exercises 1, 3, 9, 15, 19, 25, 33, 37, 39. Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 7, Section 7.2 are available on the Pearson MyLab website.

Hypothesis Testing for the Mean (σ Unknown)

Learning Objectives

After completing the readings and exercises assigned for this topic, you should be able to:

  1. Find critical values in a t distribution.
  2. Apply the t-test to test a mean μ, using the critical values/rejection region approach.

Important Note: For help accessing the eText resources referred to below, see Navigating Your eText on the course home page.

Required Reading

Elementary Statistics, Chapter 7, Section 7.3 Hypothesis Testing for the Mean (σ Unknown)

Try It Yourself Examples

Work through each Try It Yourself example in this section of the eText. Check your work against the solutions provided.

Exercises in Your eText

Do the following exercises in your eText:

Chapter 7, Section 7.3 Exercises 1, 3, 9, 11, 13,  19, 21, 25, 27. Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 7 Section 7.3 are available on the  Pearson MyLab website.

Note 1: Unless otherwise stated, always use the critical values/rejection region approach when using your calculator to work through hypotheses test exercises and problems in the Exercises sections of your textbook, in the unit Self-Test Theory Components, and Assignment Theory Components for the remainder of this course.

Note 2: Unless otherwise stated, always use the P-value approach when using your computer, with  StatCrunch, to work through hypotheses test exercises and problems in the Computer Labs, Unit Self-test Computer Components, and Assignment Computer Components for the remainder of this course.

Hypothesis Testing for Proportions

Learning Objective

After completing the readings and exercises assigned for this topic, you should be able to:

  1. Apply z in hypotheses tests involving a population proportion, using the critical values/rejection region approach.

Important Note: For help accessing the eText resources referred to below, see Navigating Your eText on the course home page.

Required Reading

Elementary Statistics, Chapter 7, Section 7.4 Hypothesis Testing for Proportions

Try It Yourself Examples

Work through each Try It Yourself example in this section of the eText. Check your work against the solutions provided.

Exercises in Your eText

Do the following exercises in your eText:

Chapter 7, Section 7.4 Exercises 1,3, 5, 7, 9, 11, 13. Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 7 Section 7.4 are available on the Pearson MyLab website.

Note: Use the critical values/rejections region approach to conduct test of hypotheses exercises, using your calculator.

Chapter 7 Review (Extra Online Practice)

For more practice working with the topics in this chapter of the eText, work through this review. Or, if you feel you have mastered this material, you may skip to Computer Lab 4B.

Review Learning Objectives

Before proceeding to the online exercises, briefly review the Learning Objectives for each of the following topics (listed below), which are presented in previous sections of this Study Guide.

  • Introduction to Hypothesis Testing with One Sample
  • Hypothesis Testing for Mean (σ Known)
  • Hypothesis Testing for Mean (σ Unknown)
  • Hypothesis Testing for Proportions
Optional Practice in the MyLab Study Plan

If you would like more practice on the various topics/sections of Chapter 7, you may wish to visit the website that accompanies your textbook, and work interactively through online exercises located in the Study Plan. For help accessing this resource, see Accessing Pearson MyLab on the course home page.

Computer Lab 4B

In Computer Lab 4B, you will learn to use StatCrunch to develop solutions to exercises related to topics in the eText’s Chapter 7.

Computer Lab 4B Detailed Instructions

Your Computer Lab activities and the detailed step-by-step instructions (Guided Solutions) that will guide you in using StatCrunch to complete these are in the Computer Lab 4B file.

Computer Lab 4B Quick Reviews

The Quick Reviews (QRs) summarize a few key steps (but not all steps) needed to complete each Activity in Computer Lab 4B. These QRs will be useful when you are preparing for the computer components of the assignments, midterm exam, and final exam. To access the QRs, click Computer Lab 4B QRs.

Self-Test 4

To access Self-Test 4, click MATH 216 Self-Test 4.

It is important that you work through all the exercises in the unit self-tests and the eText chapter quizzes. No grades are assigned to the self-tests. They are designed to, along with the unit assignments, help you master the content presented in each unit.

Each unit self-test has two parts: one on theory (A) and one on computer work (B). Working through these will help you review key exercises in the unit, which will help you prepare for assignments and exams.

Assignment 4

After completing Self-Test 4, complete Assignment 4, which you will find on the course home page. Submit your solutions to this assignment for marking using the drop box on the course home page.