Confidence Intervals for Variance and Standard Deviation 6.4 330 CHAPTER 6 Confidence Intervals What You Should Learn How to interpret the chi-square distribution and use a chi-square distribution table How to construct and interpret confidence intervals for a population variance and standard deviation The Chi-Square Distribution Confidence Intervals for s 2 and s The Chi-Square Distribution In manufacturing, it is necessary to control the amount that a process varies. For instance, an automobile part manufacturer must produce thousands of parts to be used in the manufacturing process. It is important that the parts vary little or not at all. How can you measure, and consequently control, the amount of variation in the parts? You can start with a point estimate. The point estimate for S 2 is s2 and the point estimate for S is s. The most unbiased estimate for s 2 is s2. DEFINITION You can use a chi-square distribution to construct a confidence interval for the variance and standard deviation. If a random variable x has a normal distribution, then the distribution of x 2 = 1n - 12s2 s 2 forms a chi-square distribution for samples of any size n 7 1. Here are several properties of the chi-square distribution. 1. All values of x 2 are greater than or equal to 0. 2. The chi-square distribution is a family of curves, each determined by the degrees of freedom. To form a confidence interval for s 2, use the chi-square distribution with degrees of freedom equal to one less than the sample size. d.f. = n - 1 Degrees of freedom 3. The total area under each chi-square distribution curve is equal to 1. 4. The chi-square distribution is positively skewed and therefore the distribution is not symmetric. 5. The chi-square distribution is different for each number of degrees of freedom, as shown in the figure. As the degrees of freedom increase, the chi-square distribution approaches a normal distribution. d.f. = 2 d.f. = 5 d.f. = 10 d.f. = 15 d.f. = 30 10 20 30 40 50 χ2 Chi-Square Distribution for Different Degrees of Freedom DEFINITION Study Tip The Greek letter x is pronounced “ki,” which rhymes with the more familiar Greek letter p.
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