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Mathematical Probability Ch 8 Exercises

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MAT 1600 – Chapter 8 Problem Set 1. A portfolio consists of 10,000 stocks whose returns are normally distributed. The population mean return is 12%, and the population standard deviation is 30%. A sample of 100 of these stocks is chosen; compute the following probabilities for the mean annual return in this sample: [Using Normal table attached below] a. Greater than 9% 𝑃(𝑋 ≥ 0.09) =? 𝑆𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑡𝑜 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 = 𝑛 100 = = 0.01 = 1% 𝑁 10000 Since it’s less than 5%, we conclude finite population correction factor is not applied. 𝜇𝑥̅ = 𝜇 = 0.12 𝜎𝑥̅ = 𝑍= 𝜎 √𝑛 = 0.3 √100 = 0.3 = 0.03 10 𝑥̅ − 𝜇𝑥̅ 0.09 − 0.12 = = −1 𝜎𝑥̅ 0.03 𝑃(𝑍 ≥ −1) = 1 − 𝑃(𝑍 ≤ −1) = 1 − 0.1587 = 0.8413 (Could be calculated as: 𝑛𝑜𝑟𝑚𝑎𝑙𝑐𝑑𝑓(−1, 𝐸99,0,1) = 0.8413) b. Less than 6% 𝑃(𝑋 ≤ 0.06) =? 𝑍= 𝑥̅ − 𝜇𝑥̅ 0.06 − 0.12 = = −2 𝜎𝑥̅ 0.03 𝑃(𝑍 ≤ −2) = 0.0228 (Could be calculated as: 𝑛𝑜𝑟𝑚𝑎𝑙𝑑𝑐𝑑𝑓(−𝐸99, −2,0,1) = 0.0228) c. Between 7% and 10% 𝑃(0.07 ≤ 𝑋 ≤ 0.10) =? 𝑥̅𝑙 − 𝜇𝑥̅ 0.07 − 0.12 = = −1.67 𝜎𝑥̅ 0.03 𝑥𝑢 − 𝜇𝑥̅ 0.10 − 0.12 ̅̅̅ 𝑍𝑢 = = = −0.67 𝜎𝑥̅ 0.03 𝑍𝑙 = MAT 1600 – Chapter 8 Problem Set 𝑃(−1.67 ≤ 𝑍 ≤ −0.67) = 𝑃( ...
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