Justifying a Claim Based on a Confidence Interval for a Population Mean or Population Mean Difference

Two things to interpret

A confidence interval comes from one sample, so the interval you computed may or may not contain the population mean. You never find out which. What you can say comes in two separate statements.

The interval. We are C% confident that the interval from $a$ to $b$ contains the true value of the population mean (or mean difference), described in context. For the soda line: we are 95% confident that the interval from 11.92 to 12.04 ounces contains the true mean amount of soda in all cans filled on this line.

The confidence level. The C% describes the method, not this particular interval. In repeated random sampling with the same sample size from the same population, about C% of the intervals created would capture the population mean. Any one interval either captured it or missed, and no probability is left once it has been computed.

Using an interval to judge a claim

A confidence interval is a set of plausible values for the parameter, so it can serve as evidence about a claim.

• If the claimed value is outside the interval, the interval gives convincing evidence against the claim.
• If the claimed value is inside the interval, it is plausible. The data do not give convincing evidence against it, but they do not prove it either, because other values in the interval are plausible too.

For a mean difference from paired data, the value to watch is usually 0. An interval for ${\mu }_d$ that contains 0 leaves "no difference on average" plausible. An interval that sits entirely above 0, or entirely below it, gives convincing evidence that the mean difference is not 0, and its sign tells you which direction.

What makes an interval wider or narrower

For a given sample, raising the confidence level raises the critical value $t^{\ast }$, which raises the margin of error, which widens the interval. With 25 observations, $t^{\ast }$ is 1.711 for 90% confidence, 2.064 for 95% and 2.797 for 99%. More confidence costs precision.

Raising the sample size lowers the standard error $s/\sqrt{n}$, so with everything else the same the interval tends to get narrower. The width is approximately proportional to $1/\sqrt{n}$: to cut the width in half, you need about four times as many observations. The table shows this for a sample standard deviation of 8 at 95% confidence.

Each step multiplies $n$ by 4 and leaves the width a little under half of what it was. The standard error is exactly halved; the extra shrinkage comes from $t^{\ast }$ dropping as the degrees of freedom grow, which matters most for small samples.