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Mathematical Probability Ch 9 Exercises

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MAT 1600 – Chapter 9 Problem Set 1. Assume that an investor holds a portfolio consisting of 1,000 stocks whose rates of return are normally distributed. A sample of 30 of these stocks is chosen; the sample mean returns is 5%. The sample standard deviation of the return is 8%. a. Contract a 90% confidence interval for the population mean rate of returns Since we have Sample mean and Sample Standard deviation, we use t-Test, using t-table: 90% 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 → 100(1 − 𝛼) = 90% → 𝛼 = 10% = 0.1 𝛼 0.1 = = 0.05 2 2 𝑛 = 30 𝑡 𝑛−1 𝛼 = 𝑡 29 0.05 = 1.699 2 𝑛 = 30 𝑑𝑓 = 30 − 1 = 29 Given: { 𝑠 = 0.08 𝑋̅ = 0.05 𝑋̅ ± 𝑡 𝑛−1 𝛼 2 ( 𝑠 0.08 ) = 0.05 ± (1.699) ( ) = 0.05 ± 0.0248 √𝑛 √30 = (0.0252, 0.0748) b. Construct a 95% confidence interval for the population mean rate of returns 95% 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 → 100(1 − 𝛼) = 95% → 𝛼 = 5% = 0.05 𝛼 0.05 = = 0.025 2 2 𝑛 = 30 𝑡 𝑛−1 𝛼 = 𝑡 29 0.025 = 2.045 2 𝑋̅ ± 𝑡 𝑛−1 𝛼 2 ( 𝑠 0.08 ) = 0.05 ± (2.045) ( ) = 0.05 ± 0.0299 √𝑛 √30 = (0.0201, 0.0799) c. Constructs a 99% confidence interval for the population mean rate of return 99% 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 → 100(1 − 𝛼) = 99% → 𝛼 = 1% = 0.01 𝛼 0.01 = = 0.005 2 2 𝑛 = 30 MAT 1600 – Chapter 9 Problem Set 𝑡 𝑛−1 𝛼 2 = 𝑡 29 0.005 = 2.756 𝑋̅ ± 𝑡 𝑛−1 𝛼 2 ( 𝑠 0.08 ) = 0.0 ...
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