To solve this problem, we need to state the null and alternative hypotheses and find the critical value for a left-tailed test with [tex]\(\alpha = 0.10\)[/tex] and [tex]\(n = 23\)[/tex] using [tex]\(\sigma^2 = 169\)[/tex].
### Step-by-Step Solution:
1. State the Null and Alternative Hypotheses:
- The null hypothesis ([tex]\(H_0\)[/tex]) represents the default or the status quo assumption. In this case, we assume that the true variance [tex]\(\sigma^2\)[/tex] is equal to 169.
[tex]\[
H_0: \sigma^2 = 169
\][/tex]
- The alternative hypothesis ([tex]\(H_1\)[/tex]) represents what we want to test against the null. Since this is a left-tailed test, we are testing if the variance [tex]\(\sigma^2\)[/tex] is less than 169.
[tex]\[
H_1: \sigma^2 < 169
\][/tex]
2. Determine the Degrees of Freedom:
- Degrees of freedom [tex]\(df\)[/tex] for the chi-squared distribution is calculated using the formula:
[tex]\[
df = n - 1
\][/tex]
- Given [tex]\(n = 23\)[/tex]:
[tex]\[
df = 23 - 1 = 22
\][/tex]
3. Find the Critical Value:
- For a left-tailed test, we need the critical value for the chi-squared distribution with [tex]\(22\)[/tex] degrees of freedom and [tex]\(\alpha = 0.10\)[/tex].
- The critical value ([tex]\(C.V.\)[/tex]) for this scenario is obtained from chi-squared distribution tables or by using statistical software to find the point such that the area to the left under the chi-squared curve with 22 degrees of freedom is 0.10.
Based on this information, the appropriate null and alternative hypotheses and the critical value are:
[tex]\[
\boxed{\begin{aligned}
&
H_0: \sigma^2 = 169 \\
&H_1: \sigma^2 < 169 \\
& \text{Critical Value} = 14.042
\end{aligned}}
\][/tex]
Thus, among the given options, the correct choice is:
[tex]\[
D) H_{o:} \sigma^2 = 169, H_{1:} \sigma^2 < 169, \overline{\text{C. V. }} = 14.042
\][/tex]
