95% confidence interval for the mean becomes irrelevant when t-test is insignificant?

Question

For the results I have been given, I have been asked to interpret the 95% confidence interval for the mean. However, the t-test shows a value more than .05, meaning that the differences I am exploring would be considered insignificant. Does that mean that the 95% confidence interval has any meaning or is also irrelevant? Was it a trick question?

Answer 1

That the p-value for the t-test of the hypothesis $H\!:\mu=0$ is larger than $5\%$ just means that there is not enough evidence against that hypothesis. Accepting (or rather that we fail to reject) the hypothesis $H$ does not mean that the true mean, $\mu$, is in fact $0$ but only that $0$ is among the probable values of the true mean. In fact, all values in the $95\%$ confidence interval are all probable values meaning that we do not reject the hypothesis $H\!:\mu=a$ if and only if $a$ is in the $95\%$ confidence interval.

The $95\%$ confidence interval is still very much important. The width of the interval tells you something about the power of your test - the wider the confidence interval, the lower the power.

Answer 2

No, it's not a trick question. You can always interpret the confidence interval. When you say "the $t$-test shows a value more than .05," what you really mean is that you've run a hypothesis test about some null hypothesis about the mean (presumably of the form $\mu=a$ for some number $a$), and the p-value is greater than 0.05. But all that that means is that you fail to reject the null hypothesis. In other words, it means you don't have enough statistical evidence that the null is false.

But you can still interpret the confidence interval. In fact, what the insignificance of the p-value tells you is that the confidence interval will contain the value $a$.