In 2006, 13.3\% of all live births in the United States were to mothers under 20 years of age. A sociologist claims that births to mothers under 20 years of age is decreasing. The sociologist conducts a simple random sample of 34 births and finds that 5 of them were to mothers under 20 years of age. Test the sociologist's claim at the $\alpha=0.01$ level of significance.
Preliminary:
a. Is it safe to assume that $n \leq 5 \%$ of all mothers under 20 years of age in the United States?
Yes
No
b. Verify $n p(1-p) \geq 10$. Round your answer to one decimal place.
\[
n p(1-p)=
\]
c. Since $n p(1-p)< 10$, we should use which distribution to obtain the $p$-value?
exponential distribution
uniform distribktion
binomial distribution

Step 1 ：The problem is asking us to perform a hypothesis test for a proportion. The null hypothesis is that the proportion of births to mothers under 20 years of age is 13.3%, and the alternative hypothesis is that the proportion is less than 13.3%. We are given a sample size of 34 and a sample proportion of 5/34. We are also asked to perform the test at the 0.01 level of significance.

Step 2 ：We need to check the conditions for performing a hypothesis test for a proportion. The conditions are: The sampling method is simple random sampling. The population is at least 20 times as large as the sample. The sample size is large enough such that both np and n(1-p) are greater than or equal to 10.

Step 3 ：We are told that the sampling method is simple random sampling, so the first condition is met. We are not given information about the size of the population, but we are asked to assume that the sample size is less than or equal to 5% of the population, so the second condition is met.

Step 4 ：To check the third condition, we need to calculate np and n(1-p) using the hypothesized population proportion of 13.3% (or 0.133). If both of these quantities are greater than or equal to 10, then the third condition is met and we can proceed with the hypothesis test. If not, then we cannot perform the hypothesis test using the normal distribution and will need to use a different distribution.

Step 5 ：The calculated values of np and n(1-p) are 4.522 and 29.478 respectively. Since np is less than 10, the third condition for performing a hypothesis test for a proportion is not met. Therefore, we cannot use the normal distribution for this hypothesis test. Instead, we should use the binomial distribution, which is appropriate for small sample sizes and for situations where the population proportion is close to 0 or 1.

Step 6 ：The p-value of the hypothesis test is approximately 0.706. This is greater than the significance level of 0.01, so we do not reject the null hypothesis. This means that we do not have sufficient evidence to support the sociologist's claim that the proportion of births to mothers under 20 years of age is decreasing.

Step 7 ：Final Answer: The sociologist's claim that the proportion of births to mothers under 20 years of age is decreasing is not supported by the data at the \(\alpha=0.01\) level of significance. The p-value of the hypothesis test is approximately 0.706, which is greater than the significance level of 0.01. Therefore, we do not reject the null hypothesis. This means that we do not have sufficient evidence to support the sociologist's claim. \(\boxed{0.706}\)