2.3 Mathematical and Graphical Problems 1) Think of an example involving five possible quantitative outcomes of a discrete random variable and attach a probability to each one of these outcomes. Display the outcomes, probability distribution, and cumulative probability distribution in a table. Sketch both the probability distribution and the cumulative probability distribution. Answer: Answers will vary by student. The generated table should be similar to Table 2.1 in the text, and figures should resemble Figures 2.1 and 2.2 in the text. 2) The height of male students at your college/university is normally distributed with a mean of 70 inches and a standard deviation of 3.5 inches. If you had a list of telephone numbers for male students for the purpose of conducting a survey, what would be the probability of randomly calling one of these students whose height is (a) taller than 6ʹ0ʺ? (b) between 5ʹ3ʺ and 6ʹ5ʺ? (c) shorter than 5ʹ7ʺ, the mean height of female students? (d) shorter than 5ʹ0ʺ? (e) taller than Shaq O’Neal, the center of the Miami Heat, who is 7ʹ1ʺ tall? Compare this to the probability of a woman being pregnant for 10 months (300 days), where days of pregnancy is normally distributed with a mean of 266 days and a standard deviation of 16 days. Answer: (a) Pr(Z > 0.5714) = 0.2839; (b) Pr( –2 < Z < 2) = 0.9545 or approximately 0.95; (c) Pr(Z < -0.8571) = 0.1957; (d) Pr(Z < -2.8571) = 0.0021; (e) Pr(Z > 4.2857) = 0.000009 (the text does not show values above 2.99 standard deviations, Pr(Z >2.99 = 0.0014) and Pr(Z > 2.1250) = 0.0168. 3) Calculate the following probabilities using the standard normal distribution. Sketch the probability distribution in each case, shading in the area of the calculated probability. (a) Pr(Z < 0.0) (b) Pr(Z ≤ 1.0) (c) Pr(Z > 1.96) (d) Pr(Z < –2.0) (e) Pr(Z > 1.645) (f) Pr(Z > –1.645) (g) Pr(–1.96 < Z < 1.96) (h.) Pr(Z < 2.576 or Z > 2.576) (i.) Pr(Z > z) = 0.10; find z. (j.) Pr(Z < –z or Z > z) = 0.05; find z. Answer: (a) 0.5000; (b) 0.8413; (c) 0.0250; (d) 0.0228; (e) 0.0500; (f) 0.9500; (g) 0.0500; (h) 0.0100; (i) 1.2816; (j) 1.96.