Question 6 $0 / 1$ pt $\bigcirc 2$ In 2006, $10.5 \%$ 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 44 births and finds that 4 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? binomial distribution uniform distribution exponential distribution Note: Without being given that the population is normally distributed, and that $n p(1-p)<10$, we cannot assume normal distribution and are required to use the binomial distribution. Test the claim: a. What are the null and alternative hypotheses? \[ \begin{array}{l} H_{0}: ? \vee ? \vee \square \\ H_{1}: ? \vee ? \vee \square \end{array} \] b. Based on the hypotheses, calculate the $p$-value using the binomial distribution. Round to four decimal places. \[ p \text {-value }= \] c. Make a decision. Select an answer d. Make a conclusion regarding the claim. Select an answer that the population of births to mothers under 20 years of age in the United States is decreasing. Submit Question

Step 1 ：The problem is asking us to perform a hypothesis test to determine if the proportion of births to mothers under 20 years of age is decreasing. The null hypothesis would be that the proportion is not decreasing (i.e., it is still 10.5%), and the alternative hypothesis would be that the proportion is less than 10.5%. We are given a sample size of 44 births, and 4 of them were to mothers under 20 years of age. We are also given a significance level of 0.01.

Step 2 ：First, we need to check if it is safe to assume that n is less than or equal to 5% of all mothers under 20 years of age in the United States. This is a condition for using the binomial distribution in hypothesis testing.

Step 3 ：Next, we need to verify if np(1-p) is greater than or equal to 10. This is another condition for using the binomial distribution in hypothesis testing.

Step 4 ：If both conditions are met, we can proceed with the hypothesis test using the binomial distribution. We will calculate the p-value, which is the probability of observing a result as extreme as the one in our sample, assuming the null hypothesis is true. 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 is decreasing.

Step 5 ：Based on the given data, we estimated the population size to be 880. This is a rough estimate, but it is reasonable to assume that the number of all mothers under 20 years of age in the United States is much larger than this. Therefore, it is safe to assume that n is less than or equal to 5% of the population.

Step 6 ：We also calculated np(1-p) to be 4.1349, which is less than 10. This means that we cannot use the normal approximation to the binomial distribution for this hypothesis test. Instead, we should use the exact binomial distribution.

Step 7 ：The null hypothesis is that the proportion of births to mothers under 20 years of age is 10.5%, and the alternative hypothesis is that the proportion is less than 10.5%. We will calculate the p-value using the binomial distribution.

Step 8 ：The p-value is approximately 0.5031, which is much larger than the significance level of 0.01. Therefore, we do not reject the null hypothesis. This means that we do not have enough evidence to support the sociologist's claim that the proportion of births to mothers under 20 years of age is decreasing.

Step 9 ：Final Answer: Preliminary: a. Yes b. \(n p(1-p)= 4.1\) c. binomial distribution Test the claim: a. \(H_{0}: p = 0.105, H_{1}: p < 0.105\) b. \(p \text {-value }= 0.5031\) c. Do not reject the null hypothesis. d. There is not enough evidence to support the claim that the population of births to mothers under 20 years of age in the United States is decreasing.