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 1.681 1.679 1.677 1.681 1.682 1.678 1.677 1.682 1.679 1.678 click the icon to view the chi - square distribution table. c. $h_0: σ = 0.005$ (claim); $h_a: σ ≠ 0.005$ d. $h_0: σ ≥ 0.005$; $h_a: σ < 0.005$ (claim) (b) find the critical value(s). $χ_0^2 = 3.816, 21.920$ (round to three decimal places as needed. use a comma to separate answers as needed.) identify the rejection region(s). choose the correct graph below. a. b. c. d.

Explanation:

Step1: Identify Test Type

This is a two - tailed chi - square test for standard deviation. The null hypothesis \(H_0:\sigma = 0.005\) (claim) and alternative hypothesis \(H_a:\sigma
eq0.005\). For a two - tailed test with \(\alpha = 0.05\), we split the significance level into two tails, \(\alpha/2=0.025\) in each tail. The degrees of freedom \(df=n - 1=12 - 1 = 11\).

Step2: Recall Critical Region for Two - tailed Chi - square

In a two - tailed chi - square test, the rejection regions are in both the lower tail (small values of \(\chi^{2}\)) and the upper tail (large values of \(\chi^{2}\)). The critical values are \(\chi_{1-\alpha/2}^{2}\) and \(\chi_{\alpha/2}^{2}\). For \(df = 11\) and \(\alpha=0.05\), \(\chi_{0.975}^{2}=3.816\) and \(\chi_{0.025}^{2}=21.920\). The rejection regions are \(\chi^{2}<3.816\) or \(\chi^{2}>21.920\). Looking at the graphs, Graph A has shaded regions in both the lower and upper tails, which matches the two - tailed rejection regions for a chi - square test. Graph B has a shaded region in the middle (acceptance region for two - tailed), Graph C has a shaded region only in the upper tail (one - tailed right), and Graph D has a shaded region only in the lower tail (one - tailed left).

Answer:

A