# Statistics Confidence Intervals

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.
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.