Question

the measurements of the diameters (in inches) of 12 randomly chosen golf balls are listed. at $\alpha = 0.05$, is there enough evidence to reject the claim that the standard deviation of the measurements of these diameters is 0.005? assume the population is normally distributed.
1.677 1.682 1.681
1.679 1.677 1.681
1.682 1.678 1.677
1.682 1.679 1.678
click the icon to view the chi - square distribution table.
(a) write the claim mathematically and identify $h_0$ and $h_a$. choose the correct answer below.
\\(\bigcirc\\) a. $h_0: \sigma > 0.005$; $h_a: \sigma \leq 0.005$ (claim)
\\(\bigcirc\\) b. $h_0: \sigma \leq 0.005$ (claim); $h_a: \sigma > 0.005$
\\(\bigcirc\\) c. $h_0: \sigma = 0.005$ (claim); $h_a: \sigma \
eq 0.005$
\\(\bigcirc\\) d. $h_0: \sigma \geq 0.005$; $h_a: \sigma < 0.005$ (claim)

Explanation:

1. The claim is that the standard deviation \( \sigma = 0.005 \). In hypothesis testing, the null hypothesis \( H_0 \) usually contains the claim (especially for two - tailed tests or when the claim is about a specific value).
2. The alternative hypothesis \( H_a \) is the complement of the null hypothesis when we are testing if there is enough evidence to reject the claim. Since we are testing if the standard deviation is 0.005 or not, the null hypothesis is \( H_0:\sigma = 0.005 \) (the claim) and the alternative hypothesis is \( H_a:\sigma

eq0.005 \) (a two - tailed test, as we are just testing if the standard deviation is different from 0.005, not specifically greater or less).

1. For option A, the claim is in the alternative hypothesis which is incorrect. For option B, the claim is about \( \sigma\leq0.005 \) which does not match the given claim. For option D, the claim is in the alternative hypothesis and the null hypothesis is not about the claimed value of \( \sigma = 0.005 \).

Answer:

C. \( H_0: \sigma = 0.005 \) (Claim); \( H_a: \sigma
eq 0.005 \)