Confidence interval true/false questions

Question

A sprinkler system is being installed in a large office complex. Based on a series of test runs, a 99% confidence interval for the population mean $\mu$, the average activation time of the sprinkler system (in seconds), is found to be (22, 28). Determine whether each of the following statements is true or false.

a) The 99% confidence level implies that it is quite likely that the population mean is between 22 and 28.

b) The 99% confidence level implies that it is quite likely that the sample mean is between 22 and 28.

c) The 99% confidence level implies that 99% of the sample means obtained from repeated sampling would fall between 22 and 28.

d) If a 95% confidence interval were calculated from the same data, (23, 27) would be a possible interval.

I think I have gotten a,b and d but could someone help me understand why c is true/false?

Answer 1

I don't think statement c is correct because if the confidence interval is expected to capture the true mean 99% of the time then it cannot also capture the sample means 99% of the time. The interval (22, 28) must include less than 99% of the repeated sample means. Edit after B.Liu's comment: The sample means come with extra sampling variance as compared to the population parameter (which has no variance), making things more spread out and hence "breaching" the interval we had.

Maybe this is a way to "see" it. Let's simulate 100 repeated samplings, calculate mean and 95% CI for each and plot them:

n <- 100
dat <- data.frame(
    mean= rep(NA, n),
    ci_low= rep(NA, n),
    ci_high= rep(NA, n)
)

for(i in 1:n) {
    set.seed(i)
    smp <- rnorm(n= 50, mean= 10, sd= 2) # True mean is 10
    ci <- t.test(smp)$conf.int
    dat[i, ] <- c(mean(smp), ci)
}

plot(x= 1:n, y= dat$mean, pch= 19)
segments(x0= 1, x1= 1, y0= dat$ci_low[1], dat$ci_high[1])
abline(h= dat[1,c('ci_low', 'ci_high')], col= 'blue', lty= 'dashed')

Consider the first sample (with CI marked by blue lines). If statement c were correct, you would expect ~5 of the other sample means to fall within blue lines. Clearly, more of them fall outside. (However, ~95% of the samples have confidence intervals intersecting the true mean of 10)