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I need some hints for the following questions.

Let $X_1, \dots, X_8$ be a random sample from a normal population having known mean $\mu$ and variance 2. Let $\bar X_7={1\over 7} \sum_1^7X_i$ be the average of the first 7 of them.

  1. What is the mean and variance of $X_8-\bar X_7$?

  2. If $\bar X_7=4$, give an interval that will contain the value of $X_8$ with 95% confidence.

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Some hints which might help you do the calculations:

  • The expected value of the sum of random variables is the sum of their expected values

  • The expected value of the difference of random variables is the difference of their expected values

  • The expected value of the mean of random variables is the mean of their expected values

  • The variance of the sum of independent random variables is the sum of their variances

  • The variance of the difference of independent random variables is the sum of their variances

  • The variance of the mean of independent random variables is the sum of their variances divided by the square of the number of random variables

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  • $\begingroup$ I can`t caluating course. please show me. $\endgroup$ – user72446 Dec 10 '12 at 9:07

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