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# Applied 40S April 20, 2009

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Normal approximations to binomial distributions.

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### Applied 40S April 20, 2009

1. 1. Binomial Distributions or quot;Will this SUCCEED or FAIL? 352/365: Jumpin' for Mikey by ﬂickr user Mr.Thomas
2. 2. There are ﬁve children in a family. Assume that boys and girls are equally likely. HOMEWORK 1. What is the probability that three are girls? binompdf(5, 1/2, 3) 2. What is the probability that there are at most three girls (i.e., there may be 0, 1, 2, or 3 girls)? binomcdf(5, 1/2, 3) 3. What is the probability that there are more than two girls? 1 – binomcdf(5, 1/2, 2)
3. 3. A shipment of 200 tires from a tire manufacturing company is known to include 40 defective tires. Five tires are selected at random, and each tire is replaced before the next tire is selected. (a) What is the probability of getting at most 2 defective tires? binomcdf(5, 40/200, 2) HOMEWORK (b) What is the probability of getting at least 1 defective tire? 1 - binomcdf(5, 40/200, 0) (c) What is the probability of getting 2 or 3 defective tires? binompdf(5, 40/200, 2) + binompdf(5, 40/200, 3)
4. 4. At a certain hospital, the probability that a newborn is a boy is 0.47. What is the probability that between 45 and 60 (inclusive) of the next 100 babies will be boys? HOMEWORK binomcdf(100, 0.47, 60) — binomcdf(100, 0.47, 44)
5. 5. Working With Binomial Distributions Let's take apart a typical problem about alarm clocks and see how the pieces fit together. Puzzle Alarm Clock by evadedave
6. 6. A manufacturer produces 24 yard alarms per week. Six percent of all the alarms produced are defective. What is the probability of getting two defective alarms in one week? 'S' and 'F' (Success and Failure) are the possible outcomes of a trial in a binomial experiment, and 'p' and 'q' represent the probabilities for 'S' and 'F.' • P(S) = p • P(F) = q = 1 - p • n = the number of trials • x = the number of successes in n trials • p = probability of success • q = probability of failure • P(x) = probability of getting exactly x successes in n trials Note that 'Success' in this case, is the probability of selecting a defective alarm.
7. 7. A manufacturer produces 24 yard alarms per week. Six percent of all the alarms produced are defective. What is the probability of getting two defective alarms in one week? 'S' and 'F' (Success and Failure) are the possible outcomes of a trial in a binomial experiment, and 'p' and 'q' represent the probabilities for 'S' and 'F.' • P(S) = p • P(F) = q = 1 - p So how do we answer this question? • n = the number of trials • x = the number of successes in n trials • p = probability of success • q = probability of failure • P(x) = probability of getting exactly x successes in n trials Note that 'Success' in this case, is the probability of selecting a defective alarm.
8. 8. A manufacturer produces 24 yard alarms per week. Six percent of all the alarms produced are defective. What is the probability of getting two defective alarms in one week? binompdf(trials, p, x [this is optional]) trials = number of trials p = P(success) x = speciﬁc outcome
9. 9. Now you try ... Elaine is an insurance agent. The probability that she will sell a life insurance policy to a family she visits is 0.7 (she's a really GOOD sales lady). (a) If she sees 8 families today, what is the probability that she will sell exactly 5 policies?
10. 10. Now you try ... Elaine is an insurance agent. The probability that she will sell a life insurance policy to a family she visits is 0.7 (she's a really GOOD sales lady). (b) If she sees 8 families today, what is the probability that she will sell at most 5 policies?
11. 11. The Binomial Coin Experiment http://www.math.uah.edu/stat/applets/BinomialCoinExperiment.xhtml
12. 12. Normal Approximation to the Binomial Distribution We have seen that binomial distributions and their histograms are similar to normal distributions. In certain cases, a binomial distribution is a reasonable approximation of a normal distribution. How can we tell when this is true?
13. 13. Normal Approximation to the Binomial Distribution Recall: In a normal distribution, we used values for μ and σ to solve problems, where: • μ = the population mean, and • σ = the standard deviation In a binomial distribution, we used values for 'n' and 'p' to solve problems, where: • n = number of trials, and • p = probability of success
14. 14. Normal Approximation to the Binomial Distribution We now want to use the normal Link by flickr user jontintinjordan approximation of a binomial distribution. The distribution will be approximately normal if: np ≥ 5 and nq≥ 5 th be is is Once we know that a binomial typ twe the es en LIN distribution can be approximated of the K by a normal curve we can calculate dis se tri tw the values of μ and σ like this: bu o tio ns
15. 15. Normal Approximation to An Example the Binomial Distribution Border patrol ofﬁcers estimate that 10 percent of the vehicles crossing the US - Canada border carry undeclared goods. One day the ofﬁcers searched 350 randomly selected vehicles. What is the probability that 40 or more vehicles carried undeclared goods? Is this binomial distribution approximately normal? What is n? Is np ≥ 5? What is μ? What is p? Is nq ≥ 5? What is σ? What is q?
16. 16. Are the following distributions normal approximations of binomial distributions? How do you know? (a) 60 trials where the probability (b) 60 trials where the probability of success on each trial is 0.05 of success on each trial is 0.20 (c) 600 trials where the probability (d) 80 trials where the probability of success on each trial is 0.05 of success on each trial is 0.99
17. 17. Determine the mean and standard deviation for each binomial distribution. Assume that each distribution is a reasonable HOMEWORK approximation to a normal distribution. (a) 50 trials where the probability of success for each trial is 0.35 (b) 44 trials where the probability of failure for each trial is 0.28 (c) The probability of the Espro I engine failing in less than 50 000 km is 0.08. In 1998, 16 000 engines were produced. Find the mean and standard deviation for the engines that did not fail.
18. 18. Solve the following problem using a binomial solution A laboratory supply company breeds rats for lab testing. Assume that male and female rats are equally likely to be born. HOMEWORK (a) What is the probability that of 240 animals born, exactly 110 will be female? (b) What is the probability that of 240 animals born, 110 or more will be female? (c) What is the probability that of 240 animals born, 120 or more will be female? (d) Is it correct to say that, in the above situation, P(x ≥ 120) = P(x > 119), or do we need to account for the values between 119 and 120?
19. 19. HOMEWORK The probability that a student owns a CD player is 3/5. If eight students are selected at random, what is the probability that: (a) exactly four of them own a CD player? (b) all of them own a CD player? (c) none of them own a CD player?
20. 20. HOMEWORK The probability that a motorist will use a credit card for gas purchases at a large service station on the Trans Canada Highway is 7/8. If eight cars pull up to the gas pumps, what is the probability that: (a) seven of them will use a credit card? (b) four of them will use a credit card?