Problem
Test the given claim. Assume that a simple random sample is selected from a normally distributed population. Use either the P-value method or the traditional method of testing hypotheses. Company A uses a new production method to manufacture aircraft altimeters. A simple random sample of new altimeters resulted in errors listed below. Use a 0.05 level of significance to test the claim that the new production method has errors with a standard deviation greater than $32.2 \mathrm{ft}$, which was the standard deviation for the old production method. If it appears that the standard deviation is greater, does the new production method appear to be better or worse than the old method? Should the company take any action? \[ \begin{array}{l} -44,78,-25,-75,-44,14,17,52,-9,-50,-108, \\ -108 \text { ㅁ․ } \end{array} \] What are the null and alternative hypotheses? A. $\mathrm{H}_{0}: \sigma=32.2 \mathrm{ft}$ $\mathrm{H}_{1}: \sigma>32.2 \mathrm{ft}$ B. \[ \begin{array}{l} H_{0}: \sigma=32.2 \mathrm{ft} \\ H_{1}: \sigma \neq 32.2 \mathrm{ft} \end{array} \] C. $\mathrm{H}_{0}: \sigma \neq 32.2 \mathrm{ft}$ D. $\mathrm{H}_{1}: \sigma=32.2 \mathrm{ft}$ \[ \begin{array}{l} H_{0}: \sigma>32.2 \mathrm{ft} \\ H_{1}: \sigma=32.2 \mathrm{ft} \end{array} \] E. \[ \begin{array}{l} H_{0}: \sigma<32.2 \mathrm{ft} \\ H_{1}: \sigma=32.2 \mathrm{ft} \end{array} \] F. \[ \begin{array}{l} H_{0}: \sigma=32.2 \mathrm{ft} \\ H_{1}: \sigma<32.2 \mathrm{ft} \end{array} \]
Solution
Step 1 ：The null hypothesis is usually a statement of no effect or no difference. In this case, the null hypothesis would be that the standard deviation of the new production method is equal to the standard deviation of the old production method, which is \(32.2 \mathrm{ft}\).
Step 2 ：The alternative hypothesis is what we are testing for. In this case, we are testing if the standard deviation of the new production method is greater than \(32.2 \mathrm{ft}\).
Step 3 ：Therefore, the null and alternative hypotheses are: \(H_{0}: \sigma=32.2 \mathrm{ft}\) and \(H_{1}: \sigma>32.2 \mathrm{ft}\)
Step 4 ：So, the answer is A.
Step 5 ：Final Answer: \(\boxed{A}\)
