Mathematics 216 Computer-oriented Approach to Statistics

Study Guide :: Unit 3

Probability Distributions

Introduction

In Unit 2, given a probability experiment, you assigned probabilities to various events based on probability concepts and rules. In Unit 3, you will use specific probability distributions to compute probabilities for various events.

You will begin by discussing probability distributions at a general level. A probability distribution describes a list of all possible outcomes for an experiment, along with the probabilities of each of these outcomes. Each outcome is described as specific values of a random variable. A random variable (x) represents a numerical value associated with each outcome of a probability experiment. After you construct a probability distribution, you can compute the mean or expected value, variance, and standard deviation of the probability distribution.

Once you understand the notion of a probability distribution in a general sense, you will examine discrete probability distributions. These distributions involve discrete random variables. Discrete random variables can assume only certain distinct values, typically determined through a counting process. In this unit, you will study the discrete binomial probability distribution.

The most common probability distribution that statisticians deal with is a continuous probability distribution called the normal probability distribution. Such a distribution is also called a bell curve or a mound-shaped curve, terms that describe the shape of the graphical representation of the probability distribution: a smooth, bell-shaped curve that is symmetrical around the mean of the distribution.

The exact shape of the normal curve, and therefore the probability distribution, is determined by the mean and the standard deviation of the distribution. A normal distribution with a mean of zero and a standard deviation of one unit is called a standard normal probability distribution. Any normal distribution can be transformed into a standard normal distribution using a transformation formula that converts statistical observations into the standardized values of a standard normal distribution. This transformation will enable you to compute probabilities using the Standard Normal Distribution table of probabilities (Table 4, at the back of your textbook).

When you have completed this unit, you will be ready to study topics of inferential statistics, in which you will make statements about population parameters based on sample statistics.

Probability Distributions

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:
    • discrete probability distribution; standard deviation of a discrete probability distribution
    • mean (expected value)
    • random variables; discrete random variables; continuous random variables
    • variance
  2. Given a probability experiment, construct a discrete probability distribution in table and graph format.
  3. Given a discrete probability distribution, compute the mean, variance, and standard deviation of this distribution.
  4. Compute the expected value of a discrete probability distribution. Interpret your results in terms of the context of the problem.

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 4, Section 4.1 Probability Distributions (pages 190-196)

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 4, Section 4.1 Exercises 5, 13, 15, 25, 27, 29, 31, 37 (pages 197-200). Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

Binomial Distributions

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:
    • binomial experiment
    • mean and standard deviation of a binomial distribution.
  2. Given a word problem, identify the problem as a binomial experiment.
  3. Compute binomial probabilities using binomial tables.
  4. Given a binomial probability distribution, compute the mean and standard deviation of this distribution.

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 4, Section 4.2 Binomial Distributions (pages 201-209)

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

Chapter 4, Section 4.2 Exercises 11, 13, 15, 19 (pages 210-211). 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 4 Section 4.2 are available on the textbook publisher’s MyLab website.

Chapter 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 Computer Lab 3A.

Review Learning Objectives

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

  • Probability Distributions
  • Binomial Distributions
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.

Computer Lab 3A

Computer Lab 3A Detailed Instructions

In Computer Lab  3A, you will learn to use StatCrunch to develop solutions to exercises related to topics in Chapter 4 of your eText.

For Computer Lab 3A activities, and step-by-step instructions (Guided Solutions) to familiarize you with StatCrunch, see the Computer Lab 3A file.

Computer Lab 3A Quick Reviews

The Quick Reviews (QRs) summarize a few key steps (but not all steps) needed to complete each Activity in Computer Lab 3A. 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 3A QRs.

Introduction to Normal Distributions

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:
    • continuous random variable
    • normal distribution
    • standard normal distribution
    • z-score
  2. Describe the key properties of a normal distribution.
  3. Describe the key properties of a standard normal distribution.
  4. Using standard normal distribution tables, find the numerical values for areas under the standard normal curve.
  5. Using standard normal distribution tables, find the probabilities associated with different z-score intervals. We strongly suggest that you first sketch the corresponding area under the standard normal curve, before using the standard normal distribution tables.

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 5.1 Introduction to Normal Distributions and the Standard Normal Distribution

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 5.1 Exercises 17, 19, 21, 23, 27, 31. 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 5.1 are available on the textbook publisher’s MyLab website.

Normal Distributions: Finding Probabilities

Learning Objective

After completing the readings and exercises assigned for this topic, you should be able to achieve the following learning objective.

  1. Using the standard normal distribution tables at the back of the textbook, find the probabilities associated with different x intervals for normal distributions with any mean and standard deviation. We strongly suggest that you first sketch the corresponding area under the normal curve before using the standard normal distribution tables.

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 5.2

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 5.2 Exercises 13, 15, 17, 19. 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 5.2 are available on the textbook publisher’s MyLab website.

Normal Distributions: Finding Values

Learning Objectives

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

  1. Using standard normal distribution tables (e.g., Appendix B, Table 4), find the z-scores associated with different areas under the normal curve.
  2. Using standard normal distribution tables (e.g., Appendix B in the eText), find the z-scores associated with different percentiles.
  3. Find the x-value corresponding to a given z-score.
  4. Given a normal probability for a normal distribution with any mean and standard deviation, first sketch the given area (probability) under the normal curve, and then use standard normal distribution tables (e.g., Appendix B in the eText) to find a specific data value (x-value).

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 5.3 Normal Distributions: Finding Values (pages 252-256)

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 5.3 Exercises 1, 3, 5, 9, 13, 17, 19, 21, 31, 39 (pages 257-259). 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 5.3 are available on the textbook publisher’s MyLab website.

Chapter 5 Review (Extra Online Practice)

For more practice working with the topics in Sections 1-3 of chapter 5 of the eText, work through this review. Or, if you feel you have mastered this material, you may skip Computer Lab 3B.

Review Learning Objectives

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

  • Introduction to Normal Distributions
  • Normal Distributions: Finding Probabilities
  • Normal Distributions: Finding Values
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.

Computer Lab 3B

Computer Lab 3B Detailed Instructions

In Computer Lab  3B, you will learn to use StatCrunch to develop solutions to exercises related to topics in Chapter 5 of your eText.

Your Computer Lab 3B activities, and step-by-step instructions (Guided Solutions) to familiarize you with StatCrunch, are in the Computer Lab 3B file on your course home page.

Computer Lab 3B Quick Reviews

The Quick Reviews (QRs) summarize a few key steps (but not all steps) needed to complete each Activity in Computer Lab 3B. 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 3B QRs.

Self-Test 3

To access Self-Test 3, click MATH 216 Self-Test 3.

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 3

After completing Self-Test 3, complete Assignment 3, 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.