alt="images"/> is normal with mean 280 and standard deviation
. Then the standardized
‐value from
,
is standard normal with mean 0 and standard deviation 1. From the normal table in Section 1.2, the 90th percentile of a standard normal random variable is 1.282 (since the horizontal axis value of 1.282 corresponds to an upper‐tail area of 0.1). Then
Thus, the 90th percentile of the sampling distribution of is (to the nearest ). In other words, under repeated sampling, has a distribution with an area of 0.90 to the left of (and an area of 0.10 to the right of ). This illustrates a crucial distinction between the distribution of population ‐values and the sampling distribution of —the latter is much less spread out. For example, suppose for the sake of argument that the population distribution of is normal (although this is not actually required for the central limit theorem to work). Then we can do a similar calculation to the one above to find the 90th percentile of this distribution (normal with mean 280 and standard deviation 50). In particular,
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 1.5 illustrates this point.
Figure 1.5 The 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.
1.4.2 Central limit theorem—t‐version
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.
