Question

a hospital spokesperson claims that the standard deviation of the waiting times experienced by patients in its minor emergency department is no more than 0.7 minutes. a random sample of 28 waiting times has a standard deviation of 0.8 minutes. at α = 0.10, is there enough evidence to reject the spokesperson’s claim? assume the population is normally distributed. complete parts (a) through (e) below. click the icon to view the chi-square distribution table. options for (a) (not fully shown but includes): c. $h_0: sigma = 0.7$; $h_a: sigma
eq 0.7$ (claim) d. $h_0: sigma < 0.7$ (claim); $h_a: sigma geq 0.7$ (b) find the critical value(s). $chi_0^2 = 36.741$ (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. options a, b, c, d (with chi-square distribution graphs).

Explanation:

1. Hypothesis Setup: The claim is the standard deviation is no more than 0.7, so the null hypothesis \(H_0: \sigma \leq 0.7\) (claim) and alternative \(H_a: \sigma > 0.7\). This is a right - tailed test for variance (since standard deviation test is related to variance, \(\chi^{2}=\frac{(n - 1)s^{2}}{\sigma^{2}}\)).
2. Critical Value and Rejection Region: For a right - tailed chi - square test with \(n = 28\) (so degrees of freedom \(df=n - 1=27\)) and \(\alpha = 0.10\), the critical value \(\chi_{0}^{2}\) is found from the chi - square distribution table. The rejection region for a right - tailed test is the area to the right of the critical value.

• Graph A shows a left - tailed rejection region (shaded on the left), which is for a left - tailed test (e.g., \(H_a:\sigma<\) some value).
• Graph B shows a right - tailed rejection region (shaded on the right), which matches our right - tailed test (we reject \(H_0\) when \(\chi^{2}>\chi_{0}^{2}\)).
• Graph C shows a non - rejection region in the middle (two - tailed or left - tailed non - rejection, but not our case).
• Graph D shows a two - tailed rejection region (shaded on both ends), which is for a two - tailed test (\(H_a:\sigma

eq\) some value), but our test is right - tailed. So the correct graph is B.

Answer:

B