Mathematics 215: Introduction to Statistics

Study Guide

Unit 3: Probability Distributions

In this unit, we apply the probability concepts introduced in Unit 2 to the study of probability distributions. Once you have completed this unit, you will be ready to examine topics in the field of inferential statistics. Inferential statistics, you may recall, consists of methods that use sample results to make conclusions about a population of interest.

Unit 3 begins with a discussion of random variables and types of random variables, then examines the concept of a probability distribution along with its mean and standard deviation. The remainder of Unit 3 considers specific probability distributions that you will need to know when you study topics in the field of inferential statistics in subsequent units of this course.

After completing Unit 3, you will be in a good position to compute the expected losses associated with popular games of chance, whether they be local lottery draws or the more elaborate gambling found in places like Las Vegas. As well, the theory examined will help you to appreciate the workings of the insurance industry, which charges premiums based on the probabilities of various insurable events. Finally, after you study the normal distribution in the latter part of Unit 3, you will be able to assess the probabilities of many real-world events that you might face.

Unit 3 of MATH 215 consists of the following sections:

3-1 Random Variables and Probability Distribution of a Discrete Random Variable 3-2 Mean and Standard Deviation of a Discrete Random Variable 3-3 The Binomial Probability Distribution 3-4 The Standard Normal Distribution 3-5 The Normal Distribution 3-6 The Normal Approximation to the Binomial Distribution

The unit also contains a self-test. When you have completed the material for this unit, including the self-test, complete Assignment 3.

Section 3-1: Random Variables and Probability Distribution of a Discrete Random Variable

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:
   • random variable
   • discrete random variable and continuous random variable
   • probability distribution of a discrete random variable
2. construct a probability distribution in table or graph form, given a discrete random variable defined for an experiment.
3. use a probability distribution to find the probabilities of various simple and compound events.

Reading

Read the following sections in Chapter 5 of the textbook:

• Chapter 5 Introduction
• Section 5.1
• Section 5.2

Be prepared to read the material in Chapter 5 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 5

• Overview of Some Discrete Probability Distributions (jbstatistics)

Videos Related to Section 5.1

• Random Variable Basics (Brandon Foltz)
• Discrete Random Variable Basics (Brandon Foltz)

Videos Related to Section 5.2

• Introduction to Discrete Random Variables and Discrete Probability Distributions (jbstatistics)
• “At Least” or “At Most” Probability (Brandon Myers)

Exercises

Complete the following exercises from Chapter 5 of the textbook (page numbers are for the downloadable eText):

• Exercises 5.1 and 5.3 on page 181
• Exercises 5.7 and 5.9 on page 186
• Exercise 5.13 on page 187

Remember to show your work as you develop your answers.

Solutions to these exercises are provided in the Student Solutions Manual for Chapter 5 (interactive textbook) and on page AN8 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 3-2: Mean and Standard Deviation of a Discrete Random Variable

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:
   • mean of a discrete random variable
   • standard deviation of a discrete random variable
2. compute the mean and standard deviation of a discrete random variable.

Reading

Read Section 5.3 in Chapter 5 of the textbook.

Supplementary Video Resources

These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 5.3 of the textbook.

• Expected Value and Variance of Discrete Random Variables (jbstatistics)
• Expected Value (of a Random Variable) (Brandon Foltz)
• Discrete Random Variable Variance (Brandon Foltz)

Exercises

Complete the following exercises from Chapter 5 of the textbook (page numbers are for the downloadable eText):

• Exercise 5.17 on page 192
• Exercises 5.23 and 5.25 on page 193

Solutions are provided in the Student Solutions Manual for Chapter 5 (interactive textbook) and on page AN8 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

Section 3-3: The Binomial Probability Distribution

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:
   • conditions of a binomial experiment
   • binomial distribution
   • mean and standard deviation of a binomial distribution
2. construct a binomial probability distribution, given a binomial experiment.
3. compute probabilities associated with a binomial experiment, using the binomial formula, a binomial table, or both.
4. compute the mean and standard deviation for a binomial distribution.

Reading

Read Section 5.4 in Chapter 5 of the textbook.

Note: This section completes our study of Chapter 5; we do not discuss the hypergeometric probability distribution (Section 5.5) or Poisson distributions (Section 5.6) in this course.

Supplementary Video Resources

These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 5.4 of the textbook.

• An Introduction to the Binomial Distribution (jbstatistics)
• The Binomial Distribution (Brandon Foltz)
• The Binomial Mean and Standard Deviation (Brandon Foltz)

Exercises

1. Complete the following exercises from Chapter 5 of the textbook (page numbers are for the downloadable eText):
   • Exercises 5.29, 5.31, 5.33, 5.35, 5.37, and 5.39 on pages 202–203
   • Supplementary Exercises 5.61, 5.63, 5.65, and 5.67 on page 215

2. Complete the Self-Review Test for Chapter 5 (page 217 of the downloadable eText). Omit problems 11, 12, 13, 16, 17, and 18.

   Solutions are provided in the Student Solutions Manual for Chapter 5 (interactive textbook) and on page AN9 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 found at the end of Chapter 5 (pages 215–216 of the downloadable eText).

Section 3-4: The Standard Normal Distribution

Outcomes

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

1. identify and use the two characteristics of a continuous probability distribution:
   • the probability that $x$ assumes a value within a given interval lies between 0 and 1.
   • the sum of the probabilities for the value of $x$ is equal to 1.
2. identify and use the three properties of a normal distribution:
   • the total area under the curve of a normal distribution is equal to 1.
   • the curve is symmetrical about the mean.
   • the tails of the curve extend indefinitely.
3. compute probabilities for a standard normal distribution.

Reading

Read the following sections in Chapter 6 of the textbook:

• Chapter 6 Introduction
• Section 6.1

Be prepared to read the material in Chapter 6 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 Section 6.1 of the textbook.

• An Introduction to Continuous Probability Distributions (jbstatistics)
• An Introduction to the Continuous Uniform Distribution (jbstatistics)
• An Introduction to the Normal Distribution (jbstatistics)
• A Tour of the Normal Distribution (Brandon Foltz)
  • Note:
    • When watching this video, keep in mind that in this course, you are not permitted to find probabilities using a calculator statistics package without showing the steps in the process. If you generate an answer on an exam or assignment using a statistics package and do not show all the steps, you will not receive credit.
    • In this video, pay particular attention to the diagram of a normal bell-shaped curve.
• The Standard Normal Distribution and $z\text{-scores}$ (mathisfun.com)
• Finding Areas Using the Standard Normal Table (jbstatistics)
  • Note: Finding $z\text{-scores}$ by formula and the standard normal distribution table is an important skill that will be tested on exams and assignments in this course.

Exercises

Complete Exercises 6.3, 6.11, 6.13, 6.15, and 6.17 from Chapter 6 of the textbook (page 241 of the downloadable eText).

Solutions are provided in the Student Solutions Manual for Chapter 6 (interactive textbook) and on page AN10 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

Section 3-5: The Normal Distribution

Outcomes

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

1. explain the concept of $z$ values or $z$ scores, and determine and interpret $z$ values.
2. compute probabilities for any normal distribution, given the mean and standard deviation.
3. determine the $z$ and $x$ values for a normal distribution, when an area under the normal curve is known.

Reading

Read the following sections in Chapter 6 of the textbook:

• Section 6.2
• Section 6.3
• Section 6.4

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 6.2, 6.3, and 6.4

• $z$ Scores (as a Descriptive Measure of Relative Standing) (jbstatistics)
  • Note: A normal distribution of a continuous population variable may have a mean $\mu$ which is not 0 and a standard deviation $\sigma$ from the mean which is not the unit 1. These parameters will differ depending on the variable being measured. To standardize all such cases, the standard normal distribution was created with mean 0 and standard deviation from the mean 0 to be the unit 1. This was accomplished by using mathematical rigid translations and non-rigid transformations (either contraction or expansion) of the original $x\text{-variable}$ values. This is why the $z\text{-score}$ is a measure of relative standing rather than actual standing for the $x\text{-variable}$ in question.
• $z\text{-Scores}$ (Math Meeting)
• Standardizing Normally Distributed Random Variables (jbstatistics)
• Normal Distribution Practice Problems (Jason Delaney)

Exercises

Complete the following exercises from Chapter 6 of the textbook (page numbers are for the downloadable eText):

• Exercises 6.21 and 6.23 on page 247
• Exercises 6.25 and 6.27 on page 251
• Exercises 6.31 and 6.33 on page 252
• Exercises 6.37, 6.39, 6.41, and 6.43 on page 256

Solutions are provided in the Student Solutions Manual for Chapter 6 (interactive textbook) and on page AN10 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

Section 3-6: The Normal Approximation to the Binomial Distribution

Outcomes

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

1. define, and use in context, the term “continuity correction factor.”
2. use the standard normal distribution table to approximate probabilities for binomial distributions when the sample size is very large.

Reading

Read Section 6.5 in Chapter 6 of the textbook.

Note: This section completes our study of Chapter 6; normal quantile plots (Appendix 6.1) are not covered in this course.

Supplementary Video Resources

These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 6.5 of the textbook.

• The Normal Approximation to the Binomial Distribution (jbstatistics)
• The Normal Approximation of the Binomial Distribution (searching4math)

Exercises

1. Complete the following exercises from Chapter 6 of the textbook (page numbers are for the downloadable eText):
   • Exercises 6.47, 6.49, 6.51, and 6.53 on page 262
   • Supplementary Exercises 6.55. 6.57, and 6.59 on page 264

2. Complete the Self-Review Test for Chapter 6 (pages 268–269 of the downloadable eText).

   Solutions are provided in the Student Solutions Manual for Chapter 6 (interactive textbook) and on page AN10 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

3. Complete the Unit 3 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 found at the end of Chapter 6 (pages 264–265 of the downloadable eText)

Assignment 3

Once you have completed the Unit 3 Self-Test below, complete Assignment 3. 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 3.

Unit 3 Self-Test

The self-test questions are shown here for your information. Download the Unit 3 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. Jacky’s Sportspub operates a happy hour between 4 PM and 6 PM every day. Currently, the pub is collecting data regarding the daily happy hour sales of its popular spicy beef taco appetizer plate. Over the past 40 happy hours, the pub has recorded the following daily sales of its spicy beef taco plates.

   	# of spicy beef taco plates per happy hour ($\mathit{x}$)	# of happy hours (frequency)
   	$10$	6
   	$15$	10
   	$20$	8
   	$25$	12
   	$30$	4
   Total	$100$	40

   1. Let $x$ be the number of spicy beef taco plates ordered by customers in any one happy hour period. Construct a probability distribution for $x$, based on an analysis of the past 40 happy hours as described in the table above. Keep your calculations to two decimals.
   2. Based on the probability distribution constructed in part a. above, compute the mean number of spicy beef taco plates ordered per happy hour.
   3. Based on the probability distribution constructed in part a. above, compute the standard deviation of $x$.
   4. The owners of the pub are planning to follow the strategy of using the mean computed in part b. above to help them decide how many spicy beef taco plates to serve per happy hour in the next month. Under what conditions—a large standard deviation or a small standard deviation—would they be more likely to experience severe problems relating to: either preparing insufficient taco plates to satisfy daily customer demand, or preparing far too many taco plates (resulting in significant food waste)? Explain.
2. Scientific research has established that 64% of individuals suffering migraine headaches receive instant pain-relief from taking an over-the-counter medication called Painfree. In a random sample of 8 individuals experiencing migraine headaches, use the appropriate formulae to compute the probability that
   1. exactly three individuals receive instant pain-relief from Painfree.
   2. more than six individuals receive instant pain-relief from Painfree.
   3. at most two individuals receive instant pain-relief from Painfree.
3. Circle True (T) or False (F) for each of the following:
   1. TF

      The following table, which lists all the different values of $x$, along with their respective probabilities, is a valid probability distribution.
      

      $\mathit{x}$	$\mathit{P}$($\mathit{x}$)
      $1$	$0.40$
      $2$	$0.40$
      $3$	$0.10$

   2. TF

      A binomial experiment consists of $n$ independent and identical trials where, for each trial, there are three possible outcomes.
   3. TF

      A random variable that can assume any value contained in one or more intervals is called a continuous random variable.
   4. TF

      For a continuous random variable, the probability that $x$ exceeds “$a$” is equal to the probability that $x$ is at least “$a$”.
   5. TF

      The standard normal variable is computed as follows:  $z=\left(x–\text{standard}\text{deviation}\right)⁄\text{mean}$
   6. TF

      The total area under the normal curve equals 1.
4. Protection Plus Insurance Corporation charges its policy-holders (homeowners) a $500 annual premium to be protected from the following three types of home fire damage:
   • If the homeowner experiences 100% fire damage, the company will pay out (to the homeowner) the full value of the home.
   • If the homeowner experiences 50% fire damage, the company will pay out (to the homeowner) one-half of the value of the home.
   • If the homeowner experiences 25% fire damage, the company will pay out (to the homeowner) one-quarter of the full value of the home.

   Past records indicate that the chances of the three types of fire damage for any insured home are 0.0002 for full damage, 0.0004 for 50% damage, and 0.001 for 25% damage.

   Let the random variable $x$ be the annual profit, in dollars, that Protection Plus earns from charging an annual premium of $500 for policy-holders who own a home valued at $400,000. The annual profit equals the annual premium minus the dollar amount that the insurance company pays out to cover home fire damages.

   1. Construct the probability distribution for $x$.
   2. Compute the expected value of $x$. Interpret your answer.
5. A statistics quiz consists of 20 multiple choice questions, where each question has five answer options. If a student is going to randomly guess the correct answer for each of the 20 questions, use the appropriate table from Appendix B in the eText to compute the probability that
   1. the student correctly answers exactly 5 questions.
   2. the student correctly answers at least 5 questions.
   3. the student passes the quiz.
6. The length of French fries manufactured by McCann Company is normally distributed with a mean length of 4 inches, with a standard deviation of 0.5 inches.
   1. If a McCann French fry is selected at random off the production line, find the probability that the fry
      1. will be at least 4.5 inches long.
      2. will be between 3.5 and 4.5 inches long.
      3. will be at most 4.5 inches long.
   2. McCann Company is in the process of trying to sell its fries to a very large customer, who prefers to buy longer fries, and wants to make sure that there is at least a 95% chance that a McCann fry will be longer than 3 inches. Do the fries manufactured by McCann meet this customer’s requirement?
7. Past studies indicate that 80% of all households own smartphones. A random sample of 300 households has just been selected. Compute the probability that among these 300 households selected
   1. exactly 250 households own a smartphone.
   2. at most 230 households own a smartphone.
   3. more than 250 households own a smartphone.