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 $a=0.01$ level of significance. What are the null and alternative hypotheses?Based on the hypotheses, calculate the p-value using the binomial distribution. Round to four decimal places.

Step 1 ：Define the null and alternative hypotheses. The null hypothesis is that the proportion of births to mothers under 20 years of age is 13.3%, while the alternative hypothesis is that the proportion is less than 13.3%.

Step 2 ：Calculate the p-value using the binomial distribution. The p-value is the probability of observing a result as extreme as, or more extreme than, the observed data, under the assumption that the null hypothesis is true.

Step 3 ：In this case, we observed 5 births to mothers under 20 years of age out of a sample of 34. If the null hypothesis is true, and the true proportion is 13.3%, then the probability of observing 5 or fewer such births can be calculated using the cumulative distribution function (CDF) of the binomial distribution.

Step 4 ：Using the given values, n = 34 and p = 0.133, calculate the p-value.

Step 5 ：The calculated p-value is 0.7055.

Step 6 ：Compare the p-value with the significance level of 0.01. Since the p-value is much greater than the significance level, we do not have enough evidence to reject the null hypothesis.

Step 7 ：Therefore, we cannot support the sociologist's claim that the proportion of births to mothers under 20 years of age is decreasing.

Step 8 ：Final Answer: 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%. The p-value is \(\boxed{0.7055}\).