1.4 Random Sampling
In the preceding section, we had to make some pretty restrictive assumptions (normality, known mean, known variance) in order to make statistical inferences. We now explore the connection between samples and populations a little more closely so that we can draw conclusions using fewer assumptions.
Recall that the population is the entire collection of objects under consideration, while the sample is a (random) subset of the population. Sometimes we may have a complete listing of the population (a census), but most of the time a census is too expensive and time consuming to collect. Moreover, it is seldom necessary to consider an entire population in order to make some fairly strong statistical inferences about it using just a random sample.
We are particularly interested in making statistical inferences not only about values in the population, denoted , but also about numerical summary measures such as the population mean, denoted these population summary measures are called parameters. While population parameters are unknown (in the sense that we do not have all the individual population values and so cannot calculate them), we can calculate similar quantities in the sample, such as the sample meanthese sample summary measures are called statistics. (Note the dual use of the term statistics. Up until now it has represented the notion of a general methodology for analyzing data based on probability theory, and just now it was used to represent a collection of summary measures calculated from sample data.)
However, it is not enough to just have sample statistics (such as the sample mean) that average out (over a large number of hypothetical samples) to the correct target (i.e., the population mean). We would also like sample statistics that would have low variability from one hypothetical sample to another. At the very least we need to be able to quantify this variability, known as sampling uncertainty. One way to do this is to consider the sampling distribution of a statistic, that is, the distribution of values of a statistic under repeated (hypothetical) samples. Again, we can use results from probability theory to tell us what these sampling distributions are. So, all we need to do is take a single random sample, calculate a statistic, and we will know the theoretical sampling distribution of that statistic (i.e., we will know what the statistic should average out to over repeated samples, and how much the statistic should vary over repeated samples).
1.4.1 Central limit theoremnormal version
Suppose that a random sample of data values, represented by , comes from a population that has a mean of and a standard deviation of . The sample mean, , is a pretty good estimate of the population mean, . This textbook uses for the sample mean of rather than the traditional (bar), which, in the author's experience, is unfamiliar and awkward for many students. The very famous sampling distribution of this statistic derives from the central limit theorem. This theorem states that under very general conditions, the sample mean has an approximate normal distribution with mean and standard deviation (under repeated sampling). In other words, if we were to take a large number of random samples of data values and calculate the mean for each sample, the distribution of these sample means would be a normal distribution with mean and standard deviation . Since the mean of this sampling distribution is , is an unbiased estimate of .
An amazing fact about the central limit theorem is that there is no need for the population itself to be normal (remember that we had to assume this for the calculations in Section 1.3). However, the more symmetric the distribution of the population, the better is the normal approximation for the sampling distribution of the sample mean. Also, the approximation tends to be better the larger the sample size .
So, how can we use this information? Well, the central limit theorem by itself will not help us to draw statistical inferences about the population without still having to make some restrictive assumptions. However, it is certainly a step in the right direction, so let us see what kind of calculations we can now make for the home prices example. As in Section 1.3, we will assume that and , but now we no longer need to assume that the population is normal. Imagine taking a large number of random samples of size 30 from this population and calculating the mean sale price for each sample. To get a better handle on the sampling distribution of these mean sale prices, we will find the 90th percentile of this sampling distribution. Let us do the calculation first, and then see why this might be a useful number to know.
First, we need to get some notation straight. In this section, we are not thinking about the specific sample mean we got for our actual sample of 30 sale prices, . Rather we are imagining a list of potential sample means from a population distribution with mean 280 and standard deviation 50we will call a potential sample mean in this list . From the central limit theorem, the sampling distribution of 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 uppertail 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,
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 uppertail 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 workwe 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 theoremtversion
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 tdistribution solves this problem. The density curve for a tdistribution looks very similar to a normal density curve, but the tails tend to be a little thicker, that is, tdistributions 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 tdistribution 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 tdistribution with 29 degrees of freedom: Probabilities (tail areas) and percentiles (critical values) for a tdistribution with degrees of freedom.
Compared with the corresponding table for the normal distribution in Section 1.2, the critical values are slightly larger in this table.
For example, the following table shows critical values (i.e., horizontal axis values or percentiles) and tail areas for a tdistribution with 29 degrees of freedom: Probabilities (tail areas) and percentiles (critical values) for a tdistribution with degrees of freedom.
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 tdistribution 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 calculationsthis is why the tdistribution 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 tdistribution with degrees of freedom in place of a standard normal distribution in the following tversion of the central limit theorem.