Iain Pardoe - Applied Regression Modeling

Thus, the 90th percentile of the population distribution of is (to the nearest ). This is much larger than the value we got above for the 90th percentile of the sampling distribution of ( ). This is because the sampling distribution of is less spread out than the population distribution of —the standard deviations for our example are 9.129 for the former and 50 for the latter. Figure 1.5illustrates this point.

Figure 1.5The central limit theorem in action. The upper density curve (a) shows a normal population distribution for with mean and standard deviation : the shaded area is , which lies to the right of the th percentile, . The lower density curve (b) shows a normal sampling distribution for with mean and standard deviation : the shaded area is also , which lies to the right of the th percentile, . It is not necessary for the population distribution of to be normal for the central limit theorem to work—we have used a normal population distribution here just for the sake of illustration.

We can again turn these calculations around. For example, what is the probability that is greater than 291.703? To answer this, consider the following calculation:

So, the probability that is greater than 291.703 is 0.10.

One major drawback to the normal version of the central limit theorem is that to use it we have to assume that we know the value of the population standard deviation, . A generalization of the standard normal distribution called Student's t‐distribution solves this problem. The density curve for a t‐distribution looks very similar to a normal density curve, but the tails tend to be a little “thicker,” that is, t‐distributions are a little more spread out than the normal distribution. This “extra variability” is controlled by an integer number called the degrees of freedom . The smaller this number, the more spread out the t‐distribution density curve (conversely, the higher the degrees of freedom, the more like a normal density curve it looks).

For example, the following table shows critical values (i.e., horizontal axis values or percentiles) and tail areas for a t‐distribution with 29 degrees of freedom: Probabilities (tail areas) and percentiles (critical values) for a t‐distribution with degrees of freedom.

Upper‐tail area	0.1	0.05	0.025	0.01	0.005	0.001
Critical value of	1.311	1.699	2.045	2.462	2.756	3.396
Two‐tail area	0.2	0.1	0.05	0.02	0.01	0.002

Compared with the corresponding table for the normal distribution in Section 1.2, the critical values are slightly larger in this table.

We will use the t‐distribution from this point on because it will allow us to use an estimate of the population standard deviation (rather than having to assume this value). A reasonable estimate to use is the sample standard deviation, . Since we will be using an estimate of the population standard deviation, we will be a little less certain about our probability calculations—this is why the t‐distribution needs to be a little more spread out than the normal distribution, to adjust for this extra uncertainty. This extra uncertainty will be of particular concern when we are not too sure if our sample standard deviation is a good estimate of the population standard deviation (i.e., in small samples). So, it makes sense that the degrees of freedom increases as the sample size increases. In this particular application, we will use the t‐distribution with degrees of freedom in place of a standard normal distribution in the following t‐version of the central limit theorem.