To address this hypothesis testing problem at a significance level [tex]\(\alpha = 0.001\)[/tex], let's follow the steps to find the critical value and the test statistic based on the given information.
### Given Information:
- Population mean ([tex]\(\mu\)[/tex]): 50.8
- Sample mean ([tex]\(\bar{x}\)[/tex]): 45.2
- Standard deviation ([tex]\(\sigma\)[/tex]): 12.1
- Sample size ([tex]\(n\)[/tex]): 15
- Significance level ([tex]\(\alpha\)[/tex]): 0.001
### Step 1: Determine the Critical Value
The null hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu = 50.8\)[/tex]
The alternative hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu < 50.8\)[/tex] (one-tailed test)
Since this is a left-tailed test at a significance level of [tex]\(\alpha = 0.001\)[/tex], we need to find the critical value corresponding to a cumulative probability of 0.001 under the standard normal distribution.
The critical value is:
[tex]\[
\text{critical value} = -3.09
\][/tex]
(reported to three decimal places).
### Step 2: Calculate the Test Statistic
The formula for the test statistic for a sample mean is:
[tex]\[
z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}
\][/tex]
Plug in the given values:
[tex]\[
\bar{x} = 45.2, \quad \mu = 50.8, \quad \sigma = 12.1, \quad n = 15
\][/tex]
Substitute these values into the formula:
[tex]\[
z = \frac{45.2 - 50.8}{12.1 / \sqrt{15}}
\][/tex]
First, compute the standard error ([tex]\(\sigma / \sqrt{n}\)[/tex]):
[tex]\[
\sigma / \sqrt{n} = 12.1 / \sqrt{15} \approx 3.125
\][/tex]
Now, calculate the test statistic:
[tex]\[
z = \frac{45.2 - 50.8}{3.125} \approx \frac{-5.6}{3.125} \approx -1.792
\][/tex]
So, the test statistic is:
[tex]\[
\text{test statistic} = -1.792
\][/tex]
(reported to three decimal places).
### Final Answers:
- Critical value: [tex]\(-3.09\)[/tex]
- Test statistic: [tex]\(-1.792\)[/tex]
