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AP Statistics Notes 6.3 Let T = X + Y µT = µX + µY (you can add the means of different random variables) ππ2 = ππ2 + ππ2 (you can add the variances of different random variables)* Let D = X β Y µD = µX - µY (you can subtract the means of different random variables) ππ·2 = ππ2 + ππ2 (you MUST still add the variances of different random variables)* * only true for independent random variables! 1. Using information from CCM, the number of credits X that a randomly selected full-time student is taking in the fall semester has µx = 14.65 and Οx = 2.06. The number of credits Y that a randomly selected full-time student is taking in the spring semester has µy = 13.96 and Οy = 2.00. What is the mean number of credits that a full-time student takes at CCM during the fall and spring semester? What is the standard deviation of the sum? 2. Using the information from the previous problem, what is the mean difference of the number of credits that a full-time student takes at CCM for the fall and spring semester? What is the standard deviation of the difference? This time, assume that the number of credits taken in each semester is independent. 3. To save time and money, many single people have decided to try speed dating. At a speed dating event, women sit in a circle and men spend about 5 minutes getting to know a woman before moving on to the next one. Suppose that the height M of male speed daters follows a Normal distribution with a mean of 70 inches and a standard deviation of 3.5 inches and the height F of female speed daters follows a Normal distribution with a mean of 65 inches and a standard deviation of 3 inches. What is the probability that a randomly selected male speed dater is taller than the randomly selected female speed dater with whom he is paired? 1. State: What is the probability that a randomly selected male speed dater is taller than the randomly selected female speed dater with whom he is paired? Plan: Weβll define the random variable D = M β F to represent the difference between the maleβs height and the femaleβs height. Our goal is to find P(M > F) or P(D > 0). Do: Since D is the difference of two independent Normal random variables, D follows a Normal distribution with mean οD ο½ οM ο οF = 70 β 65 = 5 inches and variance ο³ D2 ο½ ο³ M2 ο« ο³ F2 = 3.52 + 32 = 21.25. Thus, the standard deviation is ο³ D ο½ 21.25 = 4.61 inches. Thus, P(D > 0) = normalcdf(0, 99999, 5, 4.61) = 0.8610. Note: to get full credit on the AP exam when using the calculator command normalcdf, students must clearly identify the shape (Normal), center (mean = 5) and spread (standard deviation = 4.61) somewhere in their work. Conclude: There is about an 86% chance that a randomly selected male speed dater will be taller than the female he is randomly paired with. Or, in about 86% of speed dating couples, the male will be taller than the female.