Question

quarters are currently minted with weights normally distributed and having a standard deviation of 0.069. new equipment is being tested in an attempt to improve quality by reducing variation. a simple random sample of 30 quarters is obtained from those manufactured with the new equipment, and this sample has a standard deviation of 0.047. use a 0.05 significance level to test the claim that quarters manufactured with the new equipment have weights with a standard deviation less than 0.069. does the new equipment appear to be effective in reducing the variation of weights? (a) write the claim mathematically and identify $h_0$ and $h_a$. \\(\bigcirc\\) a. $h_0: \sigma \leq 0.069$ (claim); $h_a: \sigma > 0.069$ \\(\bigcirc\\) b. $h_0: \sigma \geq 0.069$ (claim); $h_a: \sigma < 0.069$ \\(\bigcirc\\) c. $h_0: \sigma \leq 0.069$; $h_a: \sigma > 0.069$ (claim) \\(\bigcirc\\) d. $h_0: \sigma \geq 0.069$; $h_a: \sigma < 0.069$ (claim) (b) find the critical value(s)

Explanation:

The claim is that the standard deviation $\sigma$ is less than 0.069. In hypothesis testing, the null hypothesis $H_0$ usually contains the equality or the opposite of the claim's direction (to set up a testable statement), and the alternative hypothesis $H_a$ contains the claim. So the claim $\sigma < 0.069$ should be in $H_a$, and $H_0$ should be $\sigma \geq 0.069$ (since we are testing if $\sigma$ is less than 0.069, the null is the complement of the alternative in terms of the inequality direction for a left - tailed test). Looking at the options, option D has $H_0:\sigma\geq0.069$ and $H_a:\sigma < 0.069$ (with the claim in $H_a$), which matches our reasoning.

Answer:

D. $H_0:\sigma\geq0.069$; $H_a:\sigma < 0.069$ (Claim)