Decide whether the normal sampling distribution can be used. If it can be used, test the claim about the population proportion p at the given level of significance $\alpha$ using the given sample statistics. Claim: $p<0.09 ; \alpha=0.05 ;$ Sample statistics: $\hat{p}=0.05, n=20$ A. \[ \begin{array}{l} H_{0}: p \leq 0.09 \\ H_{a}: p>0.09 \end{array} \] B. \[ \begin{array}{l} H_{0}: p \geq 0.09 \\ H_{a}: p<0.09 \end{array} \] C. \[ \begin{array}{l} H_{0}: p=0.09 \\ H_{a}: P \neq 0.09 \end{array} \] D. The test chonot be performed. Determine the critical value(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The critical value(s) is/are (Round to two decimal places as needed. Use a comma to separate answers as needed.) B. The test cannot be performed. Find the z-test statistic. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
Solution
Step 1 ：First, we need to check if the normal sampling distribution can be used. This can be done by checking if np and n(1-p) are both greater than 5. Here, n=20 and p=0.09. If they are both greater than 5, we can use the normal sampling distribution. If not, we cannot use it.
Step 2 ：Next, we need to set up the null and alternative hypotheses. The claim is that p<0.09, so the null hypothesis is that p>=0.09 and the alternative hypothesis is that p<0.09.
Step 3 ：After setting up the hypotheses, we can calculate the test statistic. The formula for the z-test statistic in testing a claim about a population proportion is \((\hat{p} - p) / \sqrt{(p*(1-p))/n}\), where \(\hat{p}\) is the sample proportion, p is the population proportion under the null hypothesis, and n is the sample size. In this case, \(\hat{p}=0.05, p=0.09\), and n=20.
Step 4 ：Finally, we can find the critical value(s) from the z-distribution table using the level of significance, which is 0.05 in this case. The critical value is the z-score such that the area to its right under the standard normal curve is equal to the level of significance.
Step 5 ：Calculate these values: n = 20, p = 0.09, \(\hat{p} = 0.05\), \(\alpha = 0.05\), np = 1.8, nq = 18.2, H0: p >= 0.09, Ha: p < 0.09, z = -0.625, critical_value = -1.645
Step 6 ：The normal sampling distribution cannot be used because np is less than 5. Therefore, the test cannot be performed. The null and alternative hypotheses are correctly set up as H0: p >= 0.09 and Ha: p < 0.09. The z-test statistic and critical value are not valid because the normal sampling distribution cannot be used.
Step 7 ：\(\boxed{\text{Final Answer: The test cannot be performed because the normal sampling distribution cannot be used. The null and alternative hypotheses are H0: p >= 0.09 and Ha: p < 0.09. The z-test statistic and critical value are not valid.}}\)
