An Introduction To Statistical Inference And Data Analysis

Drawing inferences about population attributes (especially means) is the primary subject of subsequent chapters, which will describe speci¯c procedures for drawing speci¯c types of inferences. However, deciding which procedure is appropriate often involves assessing the validity of certain statistical assumptions. The methods described in this chapter will be our primary tools for making such assessments.

To assess whether or not an assumption is plausible, one must be able to investigate what happens when the assumption holds. For example, if a scientist needs to decide whether or not it is plausible that her sample was drawn from a normal distribution, then she needs to be able to recognize normally distributed data. For this reason, the samples studied in this chapter were generated under carefully controlled conditions, by computer simulation. This allows us to investigate how samples drawn from speci¯ed distributions should behave, thereby providing a standard against which to compare experimental data for which the true distribution can never be known. Fortunately, S-Plus provides several convenient functions for simulating random sampling.

Example 2 Consider the experiment of tossing a fair die n = 20 times. We can simulate this experiment as follows:

>SampleSpace <- c(1,2,3,4,5,6)

>sample(x=SampleSpace,size=20,replace=T) [1] 1 6 3 2 2 3 5 3 6 4 3 2 5 3 2 2 3 2 4 2

Example 3 Consider the experiment of drawing a sample of size n = 5 from Normal(2; 3). We can simulate this experiment as follows:

> rnorm(5,mean=2,sd=sqrt(3))

[1]1.3274812 0.5901923 2.5881013 1.2222812 3.4748139

7.1The Plug-In Principle

We will employ a general methodology for relating samples to populations. In Chapters 2{6 we developed a formidable apparatus for studying populations (probability distributions). We would like to exploit this apparatus fully. Given a sample, we will pretend that the sample is a ¯nite population (discrete probability distribution) and then we will use methods for studying ¯nite populations to learn about the sample. This approach is sometimes called the Plug-In Principle.

The Plug-In Principle employs a fundamental construction:

De¯nition 7.1 Let ~x = (x1; : : : ; xn) be a sample. The empirical proba-

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bility distribution associated with ~x, denoted Pn, is the discrete probability distribution de¯ned by assigning probability 1=n to each fxig.

Notice that, if a sample contains several copies of the same numerical value, then each copy is assigned probability 1=n. This is illustrated in the following example.

Example 2 (continued) A fair die is rolled n = 20 times, resulting in the sample

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The empirical distribution P20 is the discrete distribution that assigns the following probabilities:

Notice that, although the true probabilities are P (fxig) = 1=6, the empirical

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probabilities range from :05 to :35. The fact that P20 di®ers from P is an example of sampling variation. Statistical inference is concerned with determining what the empirical distribution (the sample) tells us about the true distribution (the population).

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The empirical distribution, Pn, is an appealing way to approximate the actual probability distribution, P , from which the sample was drawn. Notice that the empirical probability of any event A is just

the observed frequency with which A occurs in the sample. By the Law of Averages, this quantity tends to the true probability of A as the size of the

observe a sample ~x = fx1; : : : ; xng. Let ¹ = EXi denote the population mean. Then

De¯nition 7.3 The plug-in estimate of ¹, denoted ¹^n, is the mean of the empirical distibution:

This quantity is called the sample mean.

=1 £ 0:05 + 2 £ 0:35 + 3 £ 0:30 + 4 £ 0:10 + 5 £ 0:10 + 6 £ 0:10

=3:15:

Notice that ¹^20 6= ¹. This is another example of sampling variation.

The variance can be estimated in the same way. Let ¾2 = Var Xi denote the population variance; then

c2

Notice that we do not refer to ¾n as the sample variance. As will be discussed in Section 8.2.2, most authors designate another, equally plausible estimate of the population variance as the sample variance.

7.3Plug-In Estimates of Quantiles

Population quantities de¯ned by quantiles can also be estimated by the plugin principle. Again, suppose that X1; : : : ; Xn » P and that we observe a sample ~x = fx1; : : : ; xng. Then

De¯nition 7.5 The plug-in estimate of a population quantile is the corresponding quantile of the empirical distribution. In particular, the sample median is the median of the empirical distribution. The sample interquartile range is the interquartile range of the empirical distribution.

Example 4 Consider the experiment of drawing a sample of size n = 20 from Uniform(1; 5). This probability distribution has a population median of 3 and a population interquartile range of 4 ¡ 2 = 2. We simulated this experiment (and listed the sample in increasing order) with the following S-Plus command:

> x <- sort(runif(20,min=1,max=5))

This resulted in the following sample:

1:124600 1:161286 1:445538 1:828181 1:853359 1:934939 1:943951 2:107977 2:372500 2:448152 2:708874 3:297806 3:418913 3:437485 3:474940 3:698471 3:740666 4:039637 4:073617 4:195613

The sample median is

2:448152 + 2:708874 = 2:578513: 2

Notice that the sample median does not exactly equal the population median. To compute the sample interquartile range, we require the ¯rst and third sample quartiles, i.e. the ® = 0:25 and ® = 0:75 sample quantiles. We must now confront the fact that De¯nition 5.5 may not specify unique quantile values. For the empirical distribution of the sample above, any number in [1:853359; 1:934939] is a sample ¯rst quartile and any number in

[3:474940; 3:698471] is a sample third quartile.

The statistical community has not agreed on a convention for resolving the ambiguity in the de¯nition of quartiles. One natural and popular possibility is to use the central value in each interval of possible quartiles. If we adopt that convention here, then the sample interquartile range is

S-Plus adopts a slightly di®erent convention. The following command

If we use these values of the ¯rst and third sample quartiles, then the sample interquartile range is 3:530823 ¡ 1:914544 = 1:616279.

Notice that the sample quantities do not exactly equal the population quantities that they estimate, regardless of which convention we adopt for de¯ning quartiles.

Used judiciously, sample quantiles can be extremely useful when trying to discern various features of the population from which the sample was drawn. The remainder of this section describes two graphical techniques for assimilating and displaying sample quantile information.

7.3.1Box Plots

Information about sample quartiles is often displayed visually, in the form of a box plot. A box plot of a sample consists of a rectangle that extends from the ¯rst to the third sample quartile, thereby drawing attention to the central 50% of the data. Thus, the length of the rectangle equals the sample interquartile range. The location of the sample median is also identi¯ed, and its location within the rectangle often provides insight into whether or not

the population from which the sample was drawn is symmetric. Whiskers extend from the ends of the rectangle, either to the extreme values of the data or to 1:5 times the sample interquartile range, whichever is less. Values that lie beyond the whiskers are called outliers and are individually identi¯ed by additional lines.

0 2 4 6 8 10

Figure 7.2: Box Plot of a Sample from Â2(3)

Example 5 The pdf of the asymmetric distribution Â2(3) was graphed in Figure 4.8. The following S-Plus commands draw a random sample of n = 100 observed values from this population, then construct a box plot of the sample:

>x <- rchisq(100,df=3)

>boxplot(x)

An example of a box plot produced by these commands is displayed in Figure 7.2. In this box plot, the numerical values in the sample are represented by the vertical axis.

The third quartile of the box plot in Figure 7.2 is farther above the median than the ¯rst quartile is below it. The short lower whisker extends

-2 0 2 4 6

Figure 7.3: Box Plots of Samples from Three Normal Distributions

quantiles of another. QQ plots are used to compare the shapes of two distributions, most commonly by plotting the observed quantiles of an empirical distribution against the corresponding quantiles of a theoretical normal distribution. In this case, a QQ plot is often called a normal probability plot. If the shape of the empirical distribution resembles a normal distribution, then the points in a normal probability plot should tend to fall on a straight line. If they do not, then we should be skeptical that the sample was drawn from a normal distribution. Extracting useful information from normal probability plots requires some practice, but the patient data analyst will be richly rewarded.

Example 4 (continued) A normal probability plot of the sample generated in Example 5 against a theoretical normal distribution is displayed in Figure 7.4. This plot was created using the following S-Plus command:

> qqnorm(x)

Notice the systematic and asymmetric bending away from linearity in this plot. In particular, the smaller quantiles are much closer to the central values