Let X have a normal distribution with mean of x andstandard deviation of x. Supp

Question

Let X have a normal distribution with mean of x andstandard deviation of x. Suppose you take a random sample size of n(n > 1) and look at the sample mean X-bar. The standarddeviation (or standard error) of X-bar is:
a. The same as the standard deviation of x. b. Greater than the standard deviation of x. c. Less than the standard deviation of x.
Lifesaver rating guaranteed w/ proper explanation. Iunderstand this material well, but welcome clarification.Thanks.
a. The same as the standard deviation of x. b. Greater than the standard deviation of x. c. Less than the standard deviation of x.
Lifesaver rating guaranteed w/ proper explanation. Iunderstand this material well, but welcome clarification.Thanks.

Explanation / Answer

C. Let X1, X2...Xn be normal variables with mean and standarddeviation . Further, let all of these variables beindependent. Thus, the standard deviation of X1 + X2 + ... Xnis sqrt(^2 + ^2 + ...^2) = *sqrt(n). However, we are looking for the mean. Thus, wedivide the sum by n. But, the standard deviation of c*X is|c| * . Therefore, the standard deviation of (X1 + X2 +...Xn) / n = * sqrt(n) / |n| = / sqrt(n) Since n is an integer greater than 1, / sqrt(n) < .

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