Probability: An Introduction with Statistical Applications, 2nd Edition

Chapter 4Functions of Random Variables; Generating Functions; Statistical Applications

4.1 Introduction

We now want to expand our applications of statistical inference first encountered in Chapter 2. In particular we want to consider tests of hypotheses and the construction of confidence intervals when continuous random variables are involved; we will also introduce simple linear regression. These considerations have direct bearing on problems of data analysis such as that encountered in the following situation.

A production process has been producing bearings with mean diameter 2.60 in.; the diameters exhibit some variability around this average value with the standard deviation of the diameters believed to be 0.03 in. A quality control inspector chooses a random sample of ten bearings and finds their average diameter to be 2.66 in. Has the process changed?

The quality control inspector here has a single observation, namely 2.66 in., the average of ten observations. This is most commonly the situation: only one sample is available; decisions must be made on the basis of that single sample. Nonetheless we can speculate on what would happen were the sampling to be repeated. In that case, another sample average will most likely occur. In order to decide whether 2.66 in. is unusual or not, we must know the probability distribution of these sample means so that the variation in the mean from sample to sample can be assessed. We can then base a test of the hypothesis that the process ...