Confidence Intervals for the Variance and Standard Deviation of a Normal Population

7.4 Confidence Intervals for the Variance and Standard Deviation of a Normal Population

  Although inferences concerning a population variance s 2 or standard deviation s are

  usually of less interest than those about a mean or proportion, there are occasions when such procedures are needed. In the case of a normal population distribution,

  inferences are based on the following result concerning the sample variance S 2 .

  THEOREM

  Let X 1 ,X 2 , c, X n

  be a random sample from a normal distribution with parameters m and s 2 . Then the rv

  has a chi-squared ( x 2 ) probability distribution with n21 df.

  As discussed in Sections 4.4 and 7.1, the chi-squared distribution is a contin- uous probability distribution with a single parameter n, called the number of degrees

  of freedom, with possible values 1, 2, 3, . . . . The graphs of several x 2 probability

  density functions (pdf’s) are illustrated in Figure 7.10. Each pdf f (x; n) is positive only for x.0 , and each has a positive skew (long upper tail), though the distribu- tion moves rightward and becomes more symmetric as n increases. To specify infer- ential procedures that use the chi-squared distribution, we need notation analogous to that for a t critical value t a,n .

  f (x; ) 8 12

  x Figure 7.10 Graphs of chi-squared density functions

  NOTATION

  Let x 2 a,n , called a chi-squared critical value, denote the number on the hori- zontal axis such that a of the area under the chi-squared curve with n df lies

  to the right of x 2 a,n .

  7.4 Confidence Intervals for the Variance and Standard Deviation of a Normal Population

  Symmetry of t distributions made it necessary to tabulate only upper-tailed t critical values ( t a,n for small values of a). The chi-squared distribution is not sym-

  metric, so Appendix Table A.7 contains values of x 2 a,n both for a near 0 and near 1, as illustrated in Figure 7.11(b). For example, x 2 and x .025,14 2 5 26.119, .95,20 (the 5th

  percentile) . 5 10.851

  Each shaded area

  2 density curve Shaded area

  Figure 7.11 x 2 a , n notation illustrated

  The rv (n 2 1)S 2 s 2 satisfies the two properties on which the general method for obtaining a CI is based: It is a function of the parameter of interest s 2 , yet its

  probability distribution (chi-squared) does not depend on this parameter. The area

  under a chi-squared curve with n df to the right of x 2 a2,n is a2 , as is the area to the left of x 2 12a2,n . Thus the area captured between these two critical values is 12a . As

  a consequence of this and the theorem just stated,

  12a2,n21 ,

  ,x

  (7.17) s 2 a2,n21 b512a

  The inequalities in (7.17) are equivalent to (n 2 1)S 2 (n 2 1)S 2 2

  x 12a2,n21

  a2,n21

  Substituting the computed value s 2 into the limits gives a CI for s 2 , and taking square

  roots gives an interval for s.

  A 100(1 2 a) confidence interval for the variance s 2 of a normal pop-

  ulation has lower limit (n 2 1)s 2 x 2 a2,n21 and upper limit (n 2 1)s 2 x 2 12a2,n21

  A confidence interval for s has lower and upper limits that are the square roots of the corresponding limits in the interval for s 2 . An upper or a lower confidence bound results from replacing a2 with a in the corresponding limit

  of the CI.

  CHAPTER 7 Statistical Intervals Based on a Single Sample