Question

the measurements of the diameters (in inches) of 12 randomly chosen golf balls are listed. at α=0.05, is there enough evidence to reject the claim that the standard deviation of the measurements of these diameters is 0.005? assume the population is normally distributed. 1.677 1.682 1. 1.679 1.677 1. 1.682 1.678 1. 1.682 1.679 1. click the icon to view the chi-square distribution table. (c) find the standardized test statistic for the $\chi^2$-test. $\chi^2 = 1.940$ (round to three decimal places as needed.) (d) decide whether to reject or fail to reject the null hypothesis. \\(\bigcirc\\) a. fail to reject \\(h_0\\). the standardized test statistic is not in the rejection region. \\(\bigcirc\\) b. fail to reject \\(h_0\\). the standardized test statistic is in the rejection region. \\(\bigcirc\\) c. reject \\(h_0\\). the standardized test statistic is in the rejection region. \\(\bigcirc\\) d. reject \\(h_0\\). the standardized test statistic is not in the rejection region.

Explanation:

1. First, recall the chi - square test for standard deviation. The null hypothesis \( H_0:\sigma = 0.005 \) and the alternative hypothesis \( H_1:\sigma

eq0.005 \) (two - tailed test). The sample size \( n = 12 \), so the degrees of freedom \( df=n - 1=11 \). For a two - tailed test with \( \alpha = 0.05 \), the critical values are \( \chi_{1-\alpha/2}^2=\chi_{0.975}^2 \) and \( \chi_{\alpha/2}^2=\chi_{0.025}^2 \). From the chi - square distribution table, \( \chi_{0.975}^2(11)=3.816 \) and \( \chi_{0.025}^2(11)=21.920 \).

1. The test statistic is \( \chi^{2}=1.940 \). Since \( 1.940<3.816 \) (the left - tailed critical value), the test statistic lies in the left - tailed rejection region. In a two - tailed chi - square test for standard deviation, if the test statistic is less than \( \chi_{1 - \alpha/2}^2 \) or greater than \( \chi_{\alpha/2}^2 \), we reject the null hypothesis. So we reject \( H_0 \) because the standardized test statistic is in the rejection region.

Answer:

C. Reject \( H_0 \). The standardized test statistic is in the rejection region.