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. Based on the hypotheses, calculate the $p$ -value using the binomial distribution.

Step 1 ：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. We are asked to test the sociologist's claim at the \(a=0.01\) level of significance. Based on the hypotheses, we need to calculate the \(p\) -value using the binomial distribution.

Step 2 ：The sociologist's claim is that the proportion of births to mothers under 20 years of age is decreasing. This is a one-tailed test because we are only interested in whether the proportion has decreased, not whether it has changed in either direction. 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\%.

Step 3 ：We can use the binomial distribution to calculate the probability of observing 5 or fewer births to mothers under 20 years of age out of a sample of 34, assuming the true proportion is 13.3\%. This probability is the \(p\)-value for the test.

Step 4 ：Given that \(n = 34\) and \(p = 0.133\), we calculate the \(p\)-value to be approximately 0.706. This is the probability of observing 5 or fewer births to mothers under 20 years of age out of a sample of 34, assuming the true proportion is 13.3\%.

Step 5 ：To test the sociologist's claim at the \(a=0.01\) level of significance, we compare the \(p\)-value to the significance level. If the \(p\)-value is less than the significance level, we reject the null hypothesis and conclude that the proportion of births to mothers under 20 years of age has decreased. If the \(p\)-value is greater than the significance level, we do not reject the null hypothesis.

Step 6 ：The \(p\)-value is approximately 0.706. Since this is greater than the significance level of 0.01, we do not reject the null hypothesis. Therefore, we do not have sufficient evidence to support the sociologist's claim that the proportion of births to mothers under 20 years of age has decreased. The final answer is \(\boxed{0.706}\).