A company manufactures precision metal bolts designed to have a diameter of 10.00mm. The quality control department periodically tests samples from the production line. A random sample of 12 bolts is measured, with the results shown below. Assume $\sigma =0.04$.

$\left(A\right)$ Use $\alpha =0.05$ to test if the mean diameter is different from 10.00 mm.
$H_0:$_________ $H_a:$_________
Because $P$-value = _______ [ < | > ] $\alpha =$ _______, we [ REJECT | FAIL TO REJECT ] $H_0$, there is [ ENOUGH | NOT ENOUGH ] evidence to suggest…

• A. Because $P$-value < $α=$ 0.05, we REJECT $H_0$, there is ENOUGH evidence to suggest μ ≠ 10.
• B. Because $P$-value < $α=$ 0.05, we REJECT $H_0$, there is ENOUGH evidence to suggest μ = 10.
• C. Because $P$-value > $α=$ 0.05, we FAIL TO REJECT $H_0$, there is NOT ENOUGH evidence to suggest μ ≠ 10.
• D. Because $P$-value > $α=$ 0.05, we FAIL TO REJECT $H_0$, there is NOT ENOUGH evidence to suggest μ = 10.

Answer: (A)

Verified step by step guidance

1. Step 1: Define the null and alternative hypotheses. Since we want to test if the mean diameter is different from 10.00 mm, set up the hypotheses as: \(H_0: \mu = 10.00\) (the mean diameter is 10.00 mm) \(H_a: \mu \neq 10.00\) (the mean diameter is not 10.00 mm).
2. Step 2: Calculate the sample mean (\(\bar{x}\)) from the given bolt diameters. Add all 12 measurements and divide by 12 to find \(\bar{x}\). Use the formula: \(\bar{x} = \frac{\sum x_i}{n}\) where \(x_i\) are the individual measurements and \(n=12\).
3. Step 3: Since the population standard deviation \(\sigma = 0.04\) is known, use the z-test for the mean. Calculate the test statistic \(z\) using the formula: \(z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}\) where \(\mu_0 = 10.00\), \(\sigma = 0.04\), and \(n=12\).
4. Step 4: Determine the critical z-value(s) for a two-tailed test at significance level \(\alpha = 0.05\). Look up the z-value that corresponds to \(\alpha/2 = 0.025\) in each tail of the standard normal distribution.
5. Step 5: Compare the calculated test statistic \(z\) to the critical z-values. Alternatively, find the p-value corresponding to the test statistic. If the p-value is less than \(\alpha = 0.05\), reject \(H_0\); otherwise, fail to reject \(H_0\). Then, state whether there is enough evidence to conclude the mean diameter differs from 10.00 mm.