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 \(\boxed{14.157}\).