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FinalProblems4A - Final Practice Problems Answer Key 1 The...

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Final Practice Problems - Answer Key 1. The following is a fictional joint probability distribution for level of household income and the type of computer a household owns. Computer type is entered in the left most column, and level of income on the top most row. Computer Type/Income 15,000 40,000 90,000 No Computer 0.2 0.1 0 PC 0.1 0.2 0.1 Mac 0 0.1 0.2 (a) What is the probability that household income is 15,000? What is the probability household income is 40,000? What is the probability that household income is 90,000? Let I denote income, C type of computer NC no computer and M denote Mac. We then obtain: P ( I = 15 , 000) = P ( I = 15 , 000; C = NC )+ P ( I = 40 , 000; C = PC )+ P ( I = 90 , 000; C = M ) = 0 . 3 Similarly, we obtain P ( I = 40 , 000) = 0 . 4 and P ( I = 90 , 000) = 0 . 3. (b) What is the probability that household income is higher than 30,000? Since P ( I 30 , 000) = P ( I = 40 , 000) + P ( I = 90 , 000) from (a) we obtain: P ( I 30 , 000) = 0 . 4 + 0 . 3 = 0 . 7 (c) Conditional on income being 90,000, what is the probability that a household owns a PC? Using our answer from (a) we quickly obtain: P ( C = PC | I = 90 , 000) = P ( C = PC ; I = 90 , 000) P ( I = 90 , 000) = 1 3 (d) Conditional on income being 90,000, what is the probability that a household owns a Mac? Following (d) we obtain P ( C = M | I = 90 , 000) = 2 / 3. (e) Are the level of income and computer ownership independent from each other? They are not. It is easy to verify, for example, by noting that: P ( I = 90 , 000; C = NC ) = 0 6 = 0 . 3 × 0 . 3 = P ( I = 90 , 000) P ( C = NC ) 2. Suppose X is normally distributed with unknown mean μ and variance σ 2 . You sample 25 observa- tions and find ¯ X = 5 and s 2 X = 16. (a) Construct a 95% confidence interval for μ using a t-distribution. What is the length of your confidence interval?
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