Section 5-1 Review and Preview

Review and Preview This chapter combines the methods of descriptive statistics presented in Chapter 2 and 3 and those of probability presented in Chapter 4 to describe and analyze probability distributions. Probability Distributions describe what will probably happen instead of what actually did happen, and they are often given in the format of a graph, table, or formula. Emphasize the combination of the methods in Chapter 3 (descriptive statistics) with the methods in Chapter 4 (probability). Chapter 3 one would conduct an actual experiment and find and observe the mean, standard deviation, variance, etc. and construct a frequency table or histogram Chapter 4 finds the probability of each possible outcome Chapter 5 presents the possible outcomes along with the relative frequencies we expect Page 200 of Elementary Statistics, 10th Edition

Preview In order to fully understand probability distributions, we must first understand the concept of a random variable, and be able to distinguish between discrete and continuous random variables. In this chapter we focus on discrete probability distributions. In particular, we discuss binomial probability distributions. Emphasize the combination of the methods in Chapter 3 (descriptive statistics) with the methods in Chapter 4 (probability). Chapter 3 one would conduct an actual experiment and find and observe the mean, standard deviation, variance, etc. and construct a frequency table or histogram Chapter 4 finds the probability of each possible outcome Chapter 5 presents the possible outcomes along with the relative frequencies we expect Page 200 of Elementary Statistics, 10th Edition

Combining Descriptive Methods and Probabilities In this chapter we will construct probability distributions by presenting possible outcomes along with the relative frequencies we expect. page 200 of Elementary Statistics, 10th Edition

Section 5-2 Random Variables

Random Variable Probability Distribution Random variable: a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure. (Example: the number of peas with green pods among 5 offspring peas.) Probability distribution: a description that gives the probability for each value of the random variable; often expressed in the format of a graph, table, or formula. page 201 of Elementary Statistics, 10th Edition

Discrete and Continuous Random Variables Discrete random variable: either a finite number of values or countable number of values, where “countable” refers to the fact that there might be infinitely many values, but they result from a counting process. (it cannot be a decimal) Continuous random variable: infinitely many values, and those values can be associated with measurements on a continuous scale without gaps or interruptions. (if something could be a decimal, it is continuous) This chapter deals exclusively with discrete random variables - experiments where the data observed is a ‘countable’ value. Give examples. Following chapters will deal with continuous random variables. Page 201 of Elementary Statistics, 10th Edition

Example 1: Identify the given random variable as being discrete or continuous. a) The number of people now driving a car in the United States. Discrete b) The weight of the gold stored in Fort Knox. Continuous c) The height of the last airplane departed from JFK Airport in New York City. page 201 of Elementary Statistics, 10th Edition

Example 1 continued: Identify the given random variable as being discrete or continuous. d) The number of cars in San Francisco that crashed last year. Discrete e) The time required to fly from Los Angeles to Shanghai. Continuous page 201 of Elementary Statistics, 10th Edition

Requirements for Probability Distribution The sum of all probabilities is 1. ΣP(x) = 1, where x assumes all possible values. (Values such as 0.999 or 1.001 are acceptable due to roundoff error) Each individual probability is a value between 0 and 1 inclusive. 0  P(x)  1, for every individual value of x. Page 203 of Elementary Statistics, 10th Edition

Standard Deviation of a Probability Distribution Mean, Variance and Standard Deviation of a Probability Distribution µ = Σ [x • P(x)] Mean σ2 = Σ[(x – µ)2 • P(x)] Variance σ2 = Σ[x2 • P(x)] – µ2 Variance (shortcut) σ = Σ[x2 • P(x)] – µ2 Standard Deviation In Chapter 3, we found the mean, standard deviation,variance, and shape of the distribution for actual observed experiments. The probability distribution and histogram can provide the same type information. These formulas will apply to ANY type of probability distribution as long as you have have all the P(x) values for the random variables in the distribution. In section 4 of this chapter, there will be special EASIER formulas for the special binomial distribution. The TI-83 and TI-83 Plus calculators can find the mean, standard deviation, and variance in the same way that one finds those values for a frequency table. With the TI-82, TI-81, and TI-85 calculators, one would have to multiply all decimal values in the P(x) column by the same factor so that there were no decimals and proceed as usual. Page 204 of Elementary Statistics, 10th Edition

Roundoff Rule for µ, , and  2 Round results by carrying one more decimal place than the number of decimal places used for the random variable x. If the values of x are integers, round µ, σ, and σ2 to one decimal place. If the rounding results in a value that looks as if the mean is ‘all’ or none’ (when, in fact, this is not true), then leave as many decimal places as necessary to correctly reflect the true mean. Page 205 of Elementary Statistics, 10th Edition

Example 2: Determine whether or not a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirement(s) that are not satisfied. Three males with X-linked genetic disorder have one child each. The random variable x is the number of children among the three who inherit the X-linked genetic disorder. The given table is a probability distribution since 0 ≤ P(x) ≤ 1 for each x and ΣP(x) = 1. x P(x) 0.125 1 0.375 2 3 page 201 of Elementary Statistics, 10th Edition

Example 3: Determine whether or not a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirement(s) that are not satisfied. Air America has a policy of routinely overbooking flights. The random variable x represents the number of passengers who cannot be boarded because there are more passengers than seats (based on data from an IBM research paper by Lawrence, Hong, and Cherrier.) The given table is not a probability distribution since ΣP(x) = 0.984 ≠ 1. x P(x) 0.051 1 0.141 2 0.274 3 0.331 4 0.187 page 201 of Elementary Statistics, 10th Edition

Example 4: Determine whether or not a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirement(s) that is/are not satisfied. The given table is not a probability distribution since ΣP(x) = 1.2 ≠ 1. x P(x) 1 0.6 2 0.2 3 4 0.15 5 0.05 page 201 of Elementary Statistics, 10th Edition

Identifying Unusual Results Range Rule of Thumb According to the range rule of thumb, most values should lie within 2 standard deviations of the mean. We can therefore identify “unusual” values by determining if they lie outside these limits: Maximum usual value = μ + 2σ Minimum usual value = μ – 2σ Page 205-206 of Elementary Statistics, 10th Edition

Identifying Unusual Results Probabilities Rare Event Rule for Inferential Statistics If, under a given assumption (such as the assumption that a coin is fair), the probability of a particular observed event (such as 992 heads in 1000 tosses of a coin) is extremely small, we conclude that the assumption is probably not correct. The discussion of this topic in the text includes some difficult concepts, but it also includes an extremely important approach used often in statistics. Page 207-208 of Elementary Statistics, 10th Edition

Identifying Unusual Results Probabilities Using Probabilities to Determine When Results Are Unusual Unusually high: x successes among n trials is an unusually high number of successes if P(x or more) ≤ 0.05. The discussion of this topic in the text includes some difficult concepts, but it also includes an extremely important approach used often in statistics. Page 207-208 of Elementary Statistics, 10th Edition Unusually low: x successes among n trials is an unusually low number of successes if P(x or fewer) ≤ 0.05.

Example 5: Refer to the table, which describes results from eight offspring peas. The random variable x represents the number of offspring peas with green pods. a) Find the probability of getting exactly 7 peas with green pods. b) Find the probability of getting 7 or more peas with green pods. x P(x) 0+ 1 2 0.004 3 0.023 4 0.087 5 0.208 6 0.311 7 0.267 8 0.100 page 201 of Elementary Statistics, 10th Edition

Example 5 continued: Refer to the table, which describes results from eight offspring peas. The random variable x represents the number of offspring peas with green pods. c) Which probability is relevant for determining whether 7 is an unusually high number of peas with green pods: the result from part (a) or part (b)? d) Is 7 an unusually high number of peas with green pods? Why or why not? x P(x) 0+ 1 2 0.004 3 0.023 4 0.087 5 0.208 6 0.311 7 0.267 8 0.100 page 201 of Elementary Statistics, 10th Edition

Example 6: Based on past results found in the Information Please Almanac, there is a 0.1919 probability that a baseball World Series contest will last four games, a 0.2121 probability that it will last five games, a 0.2222 probability that it will last six games, and a 0.3737 probability that it will last seven games. a) Does the given information describe a probability distribution? page 201 of Elementary Statistics, 10th Edition

Example 6 continued: Based on past results found in the Information Please Almanac, there is a 0.1919 probability that a baseball World Series contest will last four games, a 0.2121 probability that it will last five games, a 0.2222 probability that it will last six games, and a 0.3737 probability that it will last seven games. b) Assuming that the given information describes a probability distribution, find the mean and standard deviation for the numbers of games in World Series contests. page 201 of Elementary Statistics, 10th Edition

Example 6 continued: Based on past results found in the Information Please Almanac, there is a 0.1919 probability that a baseball World Series contest will last four games, a 0.2121 probability that it will last five games, a 0.2222 probability that it will last six games, and a 0.3737 probability that it will last seven games. c) Is it unusual for a team to “sweep” by winning in four games? Why or why not? page 201 of Elementary Statistics, 10th Edition

Example 7: Based on data from CarMax Example 7: Based on data from CarMax.com, when a car is randomly selected, the number of bumper stickers and the corresponding probabilities are as follows: 0 (0.824); 1 (0.083); 2 (0.039); 3 (0.014); 4 (0.012); 5 (0.008); 6 (0.008); 7 (0.004); 8 (0.004); 9(0.004). a) Does the given information describe a probability distribution? page 201 of Elementary Statistics, 10th Edition

Example 7 continued: Based on data from CarMax Example 7 continued: Based on data from CarMax.com, when a car is randomly selected, the number of bumper stickers and the corresponding probabilities are as follows: 0 (0.824); 1 (0.083); 2 (0.039); 3 (0.014); 4 (0.012); 5 (0.008); 6 (0.008); 7 (0.004); 8 (0.004); 9(0.004). b) Assuming that a probability distribution is described, find its mean and standard deviation. page 201 of Elementary Statistics, 10th Edition

Example 7 continued: Based on data from CarMax Example 7 continued: Based on data from CarMax.com, when a car is randomly selected, the number of bumper stickers and the corresponding probabilities are as follows: 0 (0.824); 1 (0.083); 2 (0.039); 3 (0.014); 4 (0.012); 5 (0.008); 6 (0.008); 7 (0.004); 8 (0.004); 9(0.004). c) Use the range rule of thumb to identify the range of values for usual numbers of bumper stickers. page 201 of Elementary Statistics, 10th Edition

Example 7 continued: Based on data from CarMax Example 7 continued: Based on data from CarMax.com, when a car is randomly selected, the number of bumper stickers and the corresponding probabilities are as follows: 0 (0.824); 1 (0.083); 2 (0.039); 3 (0.014); 4 (0.012); 5 (0.008); 6 (0.008); 7 (0.004); 8 (0.004); 9(0.004). d) Is it unusual for a car to have more than one bumper sticker? Why or why not? page 201 of Elementary Statistics, 10th Edition

Example 8: Let the random variable x represent the number of girls in a family of three children. Construct a table describing the probability distribution, then find the mean and standard deviation. (Hint: List the different possible outcomes) Is it unusual for a family of three children to consist of three girls?

E = Σ[x • P(x)] Expected Value The expected value of a discrete random variable is denoted by E, and it represents the mean value of the outcomes. It is obtained by finding the value of Σ [x • P(x)]. Also called expectation or mathematical expectation Plays a very important role in decision theory page 208 of Elementary Statistics, 10th Edition E = Σ[x • P(x)]

Example 9: In the Illinois Pick 3 lottery game, you pay 50¢ to select a sequence of three digits, such as 233. If you select the same sequence of three digits that are drawn, you win and collect $250. a) How many different selections are possible? b) What is the probability of winning? c) If you win, what is your net profit? page 201 of Elementary Statistics, 10th Edition

Example 9 continued: In the Illinois Pick 3 lottery game, you pay 50¢ to select a sequence of three digits, such as 233. If you select the same sequence of three digits that are drawn, you win and collect $250. d) Find the expected value. e) If you bet 50 ¢ in Illinois’ Pick 4 game, the expected value is –25¢. Which bet is better: A 50¢ bet in the Illinois Pick 3 game or a 50¢ bet in the Illinois Pick 4 game? Explain. page 201 of Elementary Statistics, 10th Edition