Sure, let's solve this step-by-step:
Step 1: State the given data.
- Population mean ([tex]\(\mu\)[/tex]) = 38 hours
- Sample size ([tex]\(n\)[/tex]) = 36
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 37.5 hours
- Standard deviation ([tex]\(s\)[/tex]) = 1.2 hours
- Significance level ([tex]\(\alpha\)[/tex]) = 0.01
Step 2: State the null and alternative hypotheses.
- Null hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu = 38\)[/tex] hours (The average assembly time hasn't changed.)
- Alternative hypothesis ([tex]\(H_1\)[/tex]): [tex]\(\mu < 38\)[/tex] hours (The average assembly time has decreased.)
Step 3: Compute the test statistic.
The test statistic for a one-sample t-test is computed using the formula:
[tex]\[ t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \][/tex]
Plugging in the given values:
[tex]\[ t = \frac{37.5 - 38}{1.2 / \sqrt{36}} = \frac{-0.5}{1.2 / 6} = \frac{-0.5}{0.2} = -2.5 \][/tex]
Step 4: Determine the degrees of freedom.
The degrees of freedom ([tex]\(df\)[/tex]) for the t-test is calculated as:
[tex]\[ df = n - 1 = 36 - 1 = 35 \][/tex]
Step 5: Find the p-value using the t-distribution.
Using the calculated t-statistic and degrees of freedom, we look up the p-value in the t-distribution table or use a statistical tool to find it.
The computed p-value for [tex]\(t = -2.5\)[/tex] with [tex]\(df = 35\)[/tex] is approximately [tex]\(0.0173\)[/tex].
Step 6: Determine the interval containing the p-value.
Given the answer choices, we place the p-value in the correct interval:
- Option a: [tex]\(0.050 < P\text{-value} < 0.075\)[/tex]
- Option b: [tex]\(0.010 < P\text{-value} < 0.020\)[/tex]
- Option c: [tex]\(0.005 < P\text{-value} < 0.010\)[/tex]
- Option d: [tex]\(0.050 < P\text{-value} < 0.100\)[/tex]
The p-value [tex]\(0.0173\)[/tex] falls within the interval [tex]\(0.010 < P\text{-value} < 0.020\)[/tex].
Conclusion:
The correct interval containing the p-value is:
[tex]\(b. \; 0.010 < P\text{-value} < 0.020\)[/tex]
