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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 β‰₯ 0.09) =? π‘†π‘Žπ‘šπ‘π‘™π‘’ 𝑠𝑖𝑧𝑒 π‘Ÿπ‘’π‘™π‘Žπ‘‘π‘–π‘£π‘’ π‘‘π‘œ π‘π‘œπ‘π‘’π‘™π‘Žπ‘‘π‘–π‘œπ‘› = 𝑛 100 = = 0.01 = 1% 𝑁 10000 Since it’s less than 5%, we conclude finite population correction factor is not applied. πœ‡π‘₯Μ… = πœ‡ = 0.12 𝜎π‘₯Μ… = 𝑍= 𝜎 βˆšπ‘› = 0.3 √100 = 0.3 = 0.03 10 π‘₯Μ… βˆ’ πœ‡π‘₯Μ… 0.09 βˆ’ 0.12 = = βˆ’1 𝜎π‘₯Μ… 0.03 𝑃(𝑍 β‰₯ βˆ’1) = 1 βˆ’ 𝑃(𝑍 ≀ βˆ’1) = 1 βˆ’ 0.1587 = 0.8413 (Could be calculated as: π‘›π‘œπ‘Ÿπ‘šπ‘Žπ‘™π‘π‘‘π‘“(βˆ’1, 𝐸99,0,1) = 0.8413) b. Less than 6% 𝑃(𝑋 ≀ 0.06) =? 𝑍= π‘₯Μ… βˆ’ πœ‡π‘₯Μ… 0.06 βˆ’ 0.12 = = βˆ’2 𝜎π‘₯Μ… 0.03 𝑃(𝑍 ≀ βˆ’2) = 0.0228 (Could be calculated as: π‘›π‘œπ‘Ÿπ‘šπ‘Žπ‘™π‘‘π‘π‘‘π‘“(βˆ’πΈ99, βˆ’2,0,1) = 0.0228) c. Between 7% and 10% 𝑃(0.07 ≀ 𝑋 ≀ 0.10) =? π‘₯̅𝑙 βˆ’ πœ‡π‘₯Μ… 0.07 βˆ’ 0.12 = = βˆ’1.67 𝜎π‘₯Μ… 0.03 π‘₯𝑒 βˆ’ πœ‡π‘₯Μ… 0.10 βˆ’ 0.12 Μ…Μ…Μ… 𝑍𝑒 = = = βˆ’0.67 𝜎π‘₯Μ… 0.03 𝑍𝑙 = MAT 1600 – Chapter 8 Problem Set 𝑃(βˆ’1.67 ≀ 𝑍 ≀ βˆ’0.67) = 𝑃( ...
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Anonymous
Awesome! Perfect study aid.

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