An article in a journal reports that $34 \%$ of American fathers take no responsibility for child care. A researcher claims that the figure is higher for fathers in the town of Littleton. A random sample of 225 fathers from Littleton, yielded 97 who did not help with child care. Find the p-value for a test of the researcher's claim. A. 0.0038 B. 0.0019 C. 0.0529 D. 0.0015

Step 1 ：An article in a journal reports that 34% of American fathers take no responsibility for child care. A researcher claims that the figure is higher for fathers in the town of Littleton. A random sample of 225 fathers from Littleton, yielded 97 who did not help with child care. We are asked to find the p-value for a test of the researcher's claim.

Step 2 ：The researcher's claim is that the proportion of fathers in Littleton who take no responsibility for child care is higher than 34%. This is a one-tailed test because we are only interested in whether the proportion is higher, not whether it is different in any direction. The null hypothesis is that the proportion is 34% and the alternative hypothesis is that the proportion is higher than 34%.

Step 3 ：We can use the formula for the test statistic in a hypothesis test for a proportion: \(Z = \frac{p - P}{\sqrt{\frac{P(1 - P)}{n}}}\) where p is the sample proportion, P is the population proportion under the null hypothesis, and n is the sample size.

Step 4 ：Substitute the given values into the formula: \(P = 0.34\), \(n = 225\), and \(p = 0.4311111111111111\).

Step 5 ：Calculate the Z score: \(Z = 2.885035594625118\).

Step 6 ：The p-value is the probability of observing a test statistic as extreme as the one we calculated, assuming the null hypothesis is true. We can find this probability using a Z-table or a statistical calculator.

Step 7 ：Calculate the p-value: \(p_{value} = 0.0019568470471456045\).

Step 8 ：The p-value calculated is approximately 0.00196. This is the probability of observing a test statistic as extreme as the one we calculated, assuming the null hypothesis is true. This p-value is less than the commonly used significance level of 0.05, which suggests that we would reject the null hypothesis and conclude that the proportion of fathers in Littleton who take no responsibility for child care is higher than 34%.

Step 9 ：Final Answer: The p-value for a test of the researcher's claim is approximately \(\boxed{0.00196}\). Looking at the options provided, the closest answer is B. 0.0019.