8-4 Testing a Claim About a Standard Deviation or Variance 417 Requirements 1. The sample is a simple random sample. 2. There is a fairly strict requirement that the population has a normal distribution. (If the second requirement is not satisfied, one alternative is to use the confidence interval method of testing hypotheses, but obtain the confidence interval using the bootstrap resampling method described in Section 8-5.) Test Statistic x2 = 1n - 12s2 s 2 (round to three decimal places, as in Table A-4) P-values: Use technology or Table A-4 with degrees of freedom: df = n - 1. See “Hint: P-Value in Two-Tailed Tests” that follows. Critical values: Use Table A-4 with degrees of freedom df = n - 1. Equivalent Methods When testing claims about s or s 2, the P-value method, the critical value method, and the confidence interval method will always lead to the same conclusion. Properties of the Chi-Square Distribution The chi-square distribution was introduced in Section 7-3, where we noted the following important properties. 1. All values of x2 are nonnegative, and the distribution is not symmetric (see Figure 8-9). 2. There is a different x2 distribution for each number of degrees of freedom (see Figure 8-10). 3. The critical values are found in Table A-4 using degrees of freedom = n − 1 Not symmetric All values are nonnegative x 2 0 FIGURE 8-9 Properties of the Chi-Square Distribution 0 5 1015202530354045 x 2 df 5 20 df 5 10 FIGURE 8-10 Chi-Square Distribution for df = 10 and df = 20 CAUTION The x2 (chi-square) test of this section is not robust against a departure from normality, meaning that the test does not work well if the population has a distribution that is far from normal. The condition of a normally distributed population is therefore a much stricter requirement when testing claims about s or s2 than when testing claims about a population mean m.
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