Question

you can calculate the p-value for a chi-square test using technology. after calculating the standardized test statistic, use the cumulative distribution function (cdf) to calculate the area under the curve. use the p-value method to test the claim.

a school administrator claims that the standard deviation for eighth-grade students on a test is greater than 35 points. a random sample of 27 eighth-grade students has a standard deviation of 36.9 points. at \\(\alpha = 0.01\\), is there enough evidence to support the administrator’s claim?

identify the null and alternative hypotheses.
\\(\bigcirc\\) a. \\(h_0: \sigma \geq 35\\)
\\(\quad\quad h_a: \sigma < 35\\)
\\(\bigcirc\\) b. \\(h_0: \sigma \leq 35\\)
\\(\quad\quad h_a: \sigma > 35\\)
\\(\bigcirc\\) c. \\(h_0: \sigma < 35\\)
\\(\quad\quad h_a: \sigma \geq 35\\)
\\(\bigcirc\\) d. \\(h_0: \sigma > 35\\)
\\(\quad\quad h_a: \sigma \leq 35\\)

Explanation:

In hypothesis testing, the null hypothesis (\(H_0\)) typically contains the equality or the opposite of the claim, and the alternative hypothesis (\(H_a\)) contains the claim. The administrator claims the standard deviation (\(\sigma\)) is greater than 35, so \(H_a: \sigma > 35\). Thus, \(H_0\) should be the complement, i.e., \(H_0: \sigma \leq 35\), which matches option B.

Answer:

B. \( H_0: \sigma \leq 35 \), \( H_a: \sigma > 35 \)