Question

a hospital spokesperson claims that the standard deviation of the waiting times experienced by patients in its emergency department is no more than 0.7 minutes. a random sample of 28 waiting times has a standard devi 0.8 minutes. at α = 0.10, is there enough evidence to reject the spokesperson’s claim? assume the population i normally distributed. complete parts (a) through (e) below. click the icon to view the chi - square distribution table. (c) find the standardized test statistic for the $\chi^2$-test. $\chi^2 = 35.265$ (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 in the rejection region. \\(\bigcirc\\) b. fail to reject \\(h_0\\). the standardized test statistic is not in the rejection region. \\(\bigcirc\\) c. reject \\(h_0\\). the standardized test statistic is not in the rejection region. \\(\bigcirc\\) d. reject \\(h_0\\). the standardized test statistic is in the rejection region.

Explanation:

1. Identify the test type: This is a chi - square test for the population variance (or standard deviation). The null hypothesis \( H_0:\sigma\leq0.7 \) (equivalent to \( \sigma^{2}\leq0.49 \)) and the alternative hypothesis \( H_a:\sigma > 0.7 \) (equivalent to \( \sigma^{2}>0.49 \)). This is a right - tailed test.
2. Determine the degrees of freedom: For a chi - square test about variance, the degrees of freedom \( df=n - 1\), where \( n = 28 \), so \( df=28 - 1=27 \).
3. Find the critical value: For a right - tailed test with \( \alpha = 0.10 \) and \( df = 27 \), from the chi - square distribution table, the critical value \( \chi_{\alpha,df}^{2}=\chi_{0.10,27}^{2}\approx36.741 \).
4. Compare test statistic and critical value: The test statistic \( \chi^{2}=35.265 \). Since \( 35.265<36.741 \), the test statistic does not fall in the rejection region (the rejection region is \( \chi^{2}>\chi_{0.10,27}^{2}\approx36.741 \)). So we fail to reject the null hypothesis.

Answer:

B. Fail to reject \( H_0 \). The standardized test statistic is not in the rejection region.