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# hw1 - Stat 704 Fall 2011 Homework 1 Due Sept 1 1 Let iid 2...

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Stat 704, Fall 2011: Homework 1 Due Sept. 1 1. Let Y 11 , Y 12 , . . . , Y 1 n 1 iid N ( μ 1 , σ 2 1 ) , independent of Y 21 , Y 22 , . . . , Y 2 n 2 iid N ( μ 2 , σ 2 2 ) . Let ¯ Y 1 = 1 n 1 n 1 i =1 Y 1 i and ¯ Y 2 = 1 n 2 n 2 i =1 Y 2 i be the sample means from the two popula- tions. (a) Find E ( ¯ Y 1 - ¯ Y 2 ). (b) Find var( ¯ Y 1 - ¯ Y 2 ). (c) What is the distribution of ¯ Y 1 - ¯ Y 2 ? Hint: first find the distributions of ¯ Y 1 and ¯ Y 2 and argue that these two random variables are independent. 2. Let Y 1 , Y 2 , and Y 3 be independent random variables with means E ( Y i ) = μ i for i = 1 , 2 , 3 and common variance var( Y i ) = σ 2 . Define ¯ Y = 1 3 ( Y 1 + Y 2 + Y 3 ). (a) Find cov( Y 1 - ¯ Y , ¯ Y ). (b) Find E { ( Y 1 + 2 Y 2 - Y 3 ) 2 } . 3. A random sample of 796 teenagers revealed that in this sample, the mean number of hours per week of TV watching was ¯ y = 13 . 2, with a standard deviation of s = 1 . 6. Find and interpret a 95% confidence interval for the true mean weekly TV-watching time for teenagers. Why can we use a t CI procedure in this problem?

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hw1 - Stat 704 Fall 2011 Homework 1 Due Sept 1 1 Let iid 2...

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