Question

use a \\(chi^2\\)-test to test the claim \\(sigma < 41\\) at the \\(alpha = 0.01\\) significance level using sample statistics \\(s = 41.8\\) and \\(n = 19\\). assume the population is normally distributed.

identify the null and alternative hypotheses.

\\(\bigcirc\\) a. \\(h_0: \sigma > 41\\)
\\(h_a: \sigma \leq 41\\)
\\(\bigcirc\\) b. \\(h_0: \sigma \leq 41\\)
\\(h_a: \sigma > 41\\)
\\(\bigcirc\\) c. \\(h_0: \sigma < 41\\)
\\(h_a: \sigma \geq 41\\)
\\(\bigcirc\\) d. \\(h_0: \sigma \geq 41\\)
\\(h_a: \sigma < 41\\)

Explanation:

To identify the null and alternative hypotheses for a hypothesis test, the null hypothesis ($H_0$) typically contains a statement of equality or no change, while the alternative hypothesis ($H_a$) contains the claim we are testing. Here, the claim is $\sigma < 41$. So, the null hypothesis should be the complement of the claim (to include equality), i.e., $H_0: \sigma \geq 41$, and the alternative hypothesis is the claim itself, $H_a: \sigma < 41$. This matches option D.

Answer:

D. $H_0: \sigma \geq 41$, $H_a: \sigma < 41$