Question

quarters are currently minted with weights normally distributed and having a standard deviation of 0.066. new equipment is being tested in an attempt to improve quality by reducing variation. a simple random sample of 28 quarters is obtained from those manufactured with the new equipment, and this sample has a standard deviation of 0.044. 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 0.066. does the new equipment appear to be effective in reducing the variation of weights? (a) write the claim mathematically and identify h₀ and hₐ. a. h₀: σ≥0.066 (claim); hₐ: σ<0.066 b. h₀: σ≤0.066; hₐ: σ>0.066 (claim) c. h₀: σ≥0.066; hₐ: σ<0.066 (claim) d. h₀: σ≤0.066 (claim); hₐ: σ>0.066

Explanation:

The claim is that the standard deviation of weights with the new equipment is less than 0.066 (\(\sigma < 0.066\)), so this is the alternative hypothesis (\(H_a\)). The null hypothesis (\(H_0\)) is the complement, which is \(\sigma \geq 0.066\). So option C correctly identifies the claim as the alternative hypothesis and sets up \(H_0\) and \(H_a\) appropriately.

Answer:

C. \( H_0: \sigma \geq 0.066 \); \( H_a: \sigma < 0.066 \) (Claim)