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True or false:

a) The center of a 95% CI for the population mean is a random variable?

d) Out of 100 95% CI for the mean $\mu$, 95 will contain the mean $\mu$?

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The assertion (a) is true. Think about how we produce a $95$ percent confidence interval. We take a random sample, with the details of size dependent on assumptions about the distribution. Then we perform certain computations to find the center of the confidence interval.

A different random sample will likely produce a somewhat different center, so the center is a function of the results in a random experiment, and is therefore a random variable.

As to (d), one should say it is false. With probability $0.95$, the CI will contain the mean $\mu$. If the confidence interval we produced contains $\mu$, call that a "success," our prediction turned out to be right.

If you repeat an experiment $100$ times, and the probability of success each time is $0.95$, that does not mean that you will necessarily get exactly $95$ successes. So (d) is false. If (d) had said "Out of $1000$ $95$% CI for $\mu$, approximately $950$ will contain the mean $\mu$", then it would be true.

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