Question

Let D represent the mean of the difference between a husband’s age and his wife’s age in Great Britain (i.e., the husband’s age - the wife’s age). We want to know if a husband’s age is greater than his wife’s age on average.

(a)_[0.5 pt] Suppose that we conduct a hypothesis test to answer the research question. Write the null and alternative hypotheses in statistical notation.

(b) Assume that the conditions for the hypothesis test are met. Use the t.test ) function in R to conduct the hypothesis test. Show your work (R code and output). Identify the test statistic value and the p-value in the output or state them.

(c) Use the t.test( ) function in R to obtain the 98% confidence interval for D, the mean of the difference between a husband’s age and his wife’s age in Great Britain (i.e., the husband’s age - the wife’s age). Assume that the conditions for the confidence interval are met.

Answer 1

The null hypothesis (H0) for the test in question is that the mean difference in age between husbands and wives in Great Britain is zero (μD = 0), and the alternative hypothesis (Ha) suggests that husbands are on average older than wives (μD > 0). A t-test can be conducted using R to find the test statistic and p-value, and the same function with a different parameter can provide a 98% confidence interval for the mean age difference.

Step-by-step explanation:

Hypothesis Testing in Statistics

To answer the research question of whether a husband's age is greater than his wife's age on average in Great Britain, we can set up a hypothesis test in statistical notation as follows:

• Null hypothesis (H0): μD = 0, which means the mean difference in age between husbands and wives (husband's age - wife's age) is zero.
• Alternative hypothesis (Ha): μD > 0, indicating that on average, husbands are older than wives, hence the mean difference is greater than zero.

To conduct a t-test in R, you can use the t.test function with sample data. Here's an example assuming age_diffs is a vector containing the differences in ages:

# R code
test_result <- t.test(age_diffs, alternative='greater', mu=0)
print(test_result)

This will output the test statistic t-value and the p-value, among other things. These are key to determining whether to reject the null hypothesis.

To find the 98% confidence interval for D using the t.test function in R, you can adjust the confidence level parameter:

# R code for 98% CI
ci_result <- t.test(age_diffs, conf.level=0.98)
print(ci_result$conf.int)

This R code will output the 98% confidence interval, which provides a range where the true mean difference is likely to fall.