STATISTICS GUIDED NOTEBOOK/FOR USE WITH MARIO TRIOLA S TEXTBOOK ESSENTIALS OF STATISTICS, 4TH ED.
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1 Example 3: There is a probability that a randomly selected 50-year old female lives through the year (based on data from the U.S. Department of Health and Human Services). A Fidelity life insurance company charges $226 for insuring that the female will live through the year. If she does not survive the year, the policy pays out $50,000 as a death benefit. a. From the perspective of the 50-year-old female, what are the values corresponding to the two events of surviving the year and not surviving? b. If a 50-year-old female purchases the policy, what is her expected value? c. Can the insurance company expect to make a profit from many such policies? Why? 5.3 BINOMIAL PROBABILITY DISTRIBUTIONS Key Concept In this section we focus on one particular category of : probability distributions. Because binomial probability distributions involve used with methods of discussed later in this book, it is important to understand properties of this particular class of probability distributions. We will present a basic definition of a probability distribution along with, and methods for finding values. probability distributions allow us to deal with circumstances CREATED BY SHANNON MARTIN GRACEY 93
2 in which the belong to relevant, such as acceptable/defective or survived/. DEFINITION A binomial probability distribution results from a procedure that meets all of the following requirements: 1. The procedure has a of trials. 2. The trials must be. 3. Each trial must have all classified into (commonly referred to as and ). 4. The probability of a remains the in all trials. NOTATION FOR BINOMIAL PROBABILITY DISTRIBUTIONS S and F (success and failure) denote the two possible categories of outcomes P S p P F 1 p q n x p q P x CREATED BY SHANNON MARTIN GRACEY 94
3 Example 1: A psychology test consists of multiple-choice questions, each having four possible answers (a, b, c, and d), one of which is correct. Assume that you guess the answers to six questions. a. Use the multiplication rule to find the probability that the first two guesses are wrong and the last four guesses are correct. b. Beginning with WWCCCC, make a complete list of the different possible arrangements of 2 wrong answers and 4 correct answers, then find the probability for each entry in the list. c. Based on the preceding results, what is the probability of getting exactly 4 correct answers when 6 guesses are made? BINOMIAL PROBABILITY FORMULA n! x n x P x p q for x 0,1, 2,, n n x! x! where n x p q CREATED BY SHANNON MARTIN GRACEY 95
4 Example 2: Assuming the probability of a pea having a green pod is 0.75, use the binomial probability formula to find the probability of getting exactly 4 peas with green pods when 5 offspring peas are generated. 5.4 MEAN, VARIANCE, AND STANDARD DEVIATION FOR THE BINOMIAL DISTRIBUTION Key Concept In this section, we consider important of distribution, including,, and. That is, given a particular binomial probability distribution, we can find its,, and. A distribution is a type of, so we could use the formulas from 5.2. However, it is easier to use the following formulas: Any Discrete pdf Binomial Distributions 1. x P x 1. np x P x x P x 2. 2 npq x P x 3. npq CREATED BY SHANNON MARTIN GRACEY 96
5 RANGE RULE OF THUMB Maximum usual value: Minimum usual value: Example 1: Mars, Inc. claims that 24% of its M&M plain candies are blue. A sample of 100 M&Ms is randomly selected. a. Find the mean and standard deviation for the numbers of blue M&Ms in such groups of 100. b. Data Set 18 in Appendix B consists of 100 M&Ms in which 27 are blue. Is this result unusual? Does it seem that the claimed rate of 24% is wrong? CREATED BY SHANNON MARTIN GRACEY 97
6 Example 2: In a study of 420,095 cell phone users in Denmark, it was found that 135 developed cancer of the brain or nervous system. If we assume that the use of cell phones has no effect on developing such cancer, then the probability of a person having such a cancer is a. Assuming that cell phones have no effect on developing cancer, find the mean and standard deviation for the numbers of people in groups of 420,095 that can be expected to have cancer of the brain or nervous system. b. Based on the results from part (a), is it unusual to find that among 420,095 people, there are 135 cases of cancer of the brain or nervous system? Why or why not? c. What do these results suggest about the publicized concern that cell phones are a health danger because they increase the risk of cancer of the brain or nervous system? CREATED BY SHANNON MARTIN GRACEY 98
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