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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?

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  • $\begingroup$ Welcome to this site! Please add the self-study tag to get more appropriate answers to your question. What bothers you about statement (c)? $\endgroup$ – chl Oct 30 at 11:02
  • $\begingroup$ Thanks! I'm just not sure about it at all. It's repeated sampling of a mean so CLT comes into play? Does the 99% CI produced for mu (22,28) hold for the repeated sampling of sample means and as such the answer would be true? I'm just struggling with teasing out what it means if that makes sense? $\endgroup$ – Laura Oct 30 at 11:08
  • $\begingroup$ Uh... Working on stats for years I still feel uncomfortable with these statements. Anyways, in my understanding the confidence interval is about the population parameter and not about the sampling statistics. Further, (c) just says "sample means" but does not specify the parameter value. Without knowing μ, we can say nothing about the probability regarding the samples. $\endgroup$ – Kota Mori Oct 30 at 11:39
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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')

enter image description here

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)

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  • 1
    $\begingroup$ I believe it will be great to also include the actual reason that "interval (22,28) must include less than 99% of the repeated sample", i.e. 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. The code and figure is great, but spelling it out in words makes it complete. $\endgroup$ – B.Liu Oct 30 at 12:13
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    $\begingroup$ @B.Liu thanks - I edited the answer with your comment. I must admit that I couldn't quite formalize myself the exact reason why it is the case. $\endgroup$ – dariober Oct 30 at 12:21

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