Sure, let's walk through solving this hypothesis testing problem together.
### Step-by-Step Solution:
Step 1: Write the Null and Alternative Hypotheses:
We are given the claim about the population variance:
- Null hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\sigma^2 \geq 8.1\)[/tex]
- Alternative hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\sigma^2 < 8.1\)[/tex]
Thus, mathematically, the hypotheses can be written as:
[tex]\[
\begin{array}{l}
H_0: \sigma^2 \geq 8.1 \\
H_a: \sigma^2 < 8.1
\end{array}
\][/tex]
Step 2: Identify the Given Data:
- Sample variance: [tex]\(s^2 = 7.97\)[/tex]
- Claimed population variance: [tex]\(\sigma^2 = 8.1\)[/tex]
- Sample size: [tex]\(n = 21\)[/tex]
- Level of significance: [tex]\(\alpha = 0.10\)[/tex]
Step 3: Calculate the Test Statistic:
For variance hypothesis tests, we use the chi-square test statistic, which is calculated using the following formula:
[tex]\[
\chi^2 = \frac{(n-1) \cdot s^2}{\sigma^2}
\][/tex]
Substituting the given values into the formula:
- [tex]\(n = 21\)[/tex]
- [tex]\(s^2 = 7.97\)[/tex]
- [tex]\(\sigma^2 = 8.1\)[/tex]
[tex]\[
\chi^2 = \frac{(21-1) \cdot 7.97}{8.1}
\][/tex]
Calculating the numerator:
[tex]\[
(21-1) \cdot 7.97 = 20 \cdot 7.97 = 159.4
\][/tex]
Calculating the chi-square statistic:
[tex]\[
\chi^2 = \frac{159.4}{8.1} \approx 19.68
\][/tex]
Step 4: Final Test Statistic:
Thus, the standardized test statistic is:
[tex]\[
\chi^2 = 19.68
\][/tex]
This concludes the step-by-step calculation of the test statistic for the given hypothesis test problem.
