Iain Pardoe - Applied Regression Modeling

Suppose that a random sample of data values, represented by , comes from a population that has a mean of . Imagine taking a large number of random samples of data values and calculating the mean and standard deviation for each sample. As before, we will let represent the imagined list of repeated sample means, and similarly, we will let represent the imagined list of repeated sample standard deviations. Define

Under very general conditions, t has an approximate t‐distribution with degrees of freedom. The two differences from the normal version of the central limit theorem that we used before are that the repeated sample standard deviations, , replace an assumed population standard deviation, , and that the resulting sampling distribution is a t‐distribution (not a normal distribution).

To illustrate, let us repeat the calculations from Section 1.4.1based on an assumed population mean, , but rather than using an assumed population standard deviation, , we will instead use our observed sample standard deviation, 53.8656 for . To find the 90th percentile of the sampling distribution of the mean sale price, :

Thus, the 90th percentile of the sampling distribution of is (to the nearest ).

Turning this around, what is the probability that is greater than 292.893?

So, the probability that is greater than 292.893 is 0.10.

So far, we have focused on the sampling distribution of sample means, , but what we would really like to do is infer what the observed sample mean, , tells us about the population mean, . Thus, while the preceding calculations have been useful for building up intuition about sampling distributions and manipulating probability statements, their main purpose has been to prepare the ground for the next two sections, which cover how to make statistical inferences about the population mean, .

We have already seen that the sample mean, , is a good point estimate of the population mean, (in the sense that it is unbiased—see Section 1.4). It is also helpful to know how reliable this estimate is, that is, how much sampling uncertainty is associated with it. A useful way to express this uncertainty is to calculate an interval estimate or confidence interval for the population mean, . The interval should be centered at the point estimate (in this case, ), and since we are probably equally uncertain that the population mean could be lower or higher than this estimate, it should have the same amount of uncertainty either side of the point estimate. We quantify this uncertainty with a number called the “margin of error.” Thus, the confidence interval is of the form “point estimate margin of error” or “(point estimate margin of error, point estimate margin of error).”

We can obtain the exact form of the confidence interval from the t‐version of the central limit theorem, where has an approximate t‐distribution with degrees of freedom. In particular, suppose that we want to calculate a 95% confidence interval for the population mean, , for the home prices example—in other words, an interval such that there will be an area of 0.95 between the two endpoints of the interval (and an area of 0.025 to the left of the interval in the lower tail, and an area of 0.025 to the right of the interval in the upper tail). Let us consider just one side of the interval first. Since 2.045 is the 97.5th percentile of the t‐distribution with 29 degrees of freedom (see the t‐table in Section 1.4.2), then