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 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. (a) write the claim mathematically and identify h₀ and hₐ. a. h₀: σ ≤ 0.7 (claim); hₐ: σ > 0.7 b. h₀: σ ≥ 0.7; hₐ: σ < 0.7 (claim) c. h₀: σ = 0.7; hₐ: σ ≠ 0.7 (claim) d. h₀: σ < 0.7 (claim); hₐ ≥ 0.7 (b) find the critical value(s). χ₀² = \boxed{} (round to three decimal places as needed. use a comma to separate answers as needed.)

Explanation:

Part (a)

The hospital spokesperson claims the standard deviation (\(\sigma\)) is no more than 0.7 minutes, so the claim is \( \sigma \leq 0.7 \), which becomes the null hypothesis (\(H_0\)). The alternative hypothesis (\(H_a\)) is the opposite of the null for a right - tailed test (since we are testing if there's evidence to reject the claim that \(\sigma\leq0.7\), we check if \(\sigma > 0.7\)).

Part (b)

Step 1: Determine the degrees of freedom

The sample size \(n = 28\). For a chi - square test of variance (or standard deviation), the degrees of freedom \(df=n - 1\). So, \(df=28 - 1=27\).

Step 2: Determine the significance level and the type of test

The significance level \(\alpha = 0.10\), and since the alternative hypothesis is \(H_a:\sigma>0.7\) (a right - tailed test), we need to find the critical value \(\chi_{\alpha,df}^{2}\).
We look up the chi - square distribution table for \(df = 27\) and \(\alpha=0.10\). From the chi - square distribution table, \(\chi_{0.10,27}^{2}=36.741\) (using a chi - square table or statistical software to find the value for a right - tailed test with \(df = 27\) and \(\alpha = 0.10\)).

Answer:

A. \( H_0: \sigma \leq 0.7 \) (Claim); \( H_a: \sigma > 0.7 \)