Data Analysis with Microsoft Excel: Updated for Office 2007

Chapter 6 Statistical Inference 225

T

he concepts you learned in Chapter 5 provide the basis for the subject

of this chapter, statistical inference. Two of the main tools of statistical

inference are confi dence intervals and hypothesis tests. In this chapter,

you’ll apply these tools to reach conclusions about your data. You’ll be

introduced to a new distribution, the t distribution, and you’ll see how to use

it in performing statistical inference. You’ll also learn about nonparametric

tests that make fewer assumptions about the distribution of your data.

Confi dence Intervals

In the previous chapter, you learned two very important facts about distri-

butions and samples.

1. A sample average will approximately follow a normal distribution with
   mean m and standard deviation s/!n, where m is the mean of the proba-
   bility distribution the sample is drawn from, s is the standard deviation
   of the probability distribution, and n is the size of the sample. Another
   way of writing this is

x,Nam, s/!nb

2. In a normal distribution, about 95% of the time, the values fall within 2
   standard deviations of the mean.
   From these two facts, we can calculate how precisely the sample average
   estimates the value of m. For example, if s 510 and our sample size is 25, the
   sample average will approximately follow a normal distribution with mean
   m and standard deviation 2, so 95% of the time, the sample average will fall
   within 4 units of m. This indicates that if the sample average is 20, we could
   construct a confi dence interval from about 16 to 24 that should, with 95%
   confi dence, “capture” the value of m. If we want this confi dence interval to
   be smaller, we simply increase the sample size. A sample of 100 observations
   would result in a 95% confi dence interval for m ranging from about 18 to 22.
   The use of the 2 standard deviations rule is an approximation. What if we
   wanted a more exact estimate of the 95% confi dence interval, or what if we
   wanted to construct other confi dence intervals, such as a 99% confi dence
   interval? How would we go about doing that?

z Test Statistic and z Values

In order to derive a more general expression of the confi dence interval, we

fi rst have to express the sample average in terms of a standard normal dis-

tribution. We can do this by subtracting the value of m and dividing by the