Use a $\chi^{2}$-test to test the claim $\sigma< 38$ at the $\alpha=0.10$ significance level using sample statistics $s=33.7$ and $n=19$. Assume the population is normally distributed.

Identify the null and alternative hypotheses
A.
\[
\begin{array}{l}
H_{0} . \sigma< 38 \\
H_{a} . \sigma \geq 38
\end{array}
\]
c.
\[
\begin{array}{l}
H_{0} ; \sigma> 38 \\
H_{a} ; \\
\sigma
\end{array}
\]
B.
\[
\begin{array}{l}
H_{0}, \sigma \geq 38 \\
H_{a}, \sigma< 38
\end{array}
\]
D.
\[
\begin{array}{l}
H_{0} \sigma \leq 38 \\
H_{2}: \sigma> 38
\end{array}
\]

Identify the standardized test statistic
(Round to three decimal places as needed)

Step 1 ：Identify the null and alternative hypotheses. The null hypothesis \(H_{0}\) is that the population standard deviation \(\sigma\) is greater than or equal to 38. The alternative hypothesis \(H_{a}\) is that the population standard deviation \(\sigma\) is less than 38.

Step 2 ：Calculate the standardized test statistic using the formula \(\chi^{2} = \frac{(n-1) \cdot s^{2}}{\sigma^{2}}\), where \(n\) is the sample size, \(s\) is the sample standard deviation, and \(\sigma\) is the population standard deviation. Substituting the given values, we get \(\chi^{2} = \frac{(19-1) \cdot 33.7^{2}}{38^{2}}\).

Step 3 ：Calculate the value of the standardized test statistic. The calculated chi-square value is 14.157 (rounded to three decimal places).

Step 4 ：The final answer is the standardized test statistic, which is \(\boxed{14.157}\).