Age at First Marriage A marriage counselor claims that the median age of men at the time of their first marriage is greater than 28 years old. In a random sample of 56 men, 33 were less than 28 years old and 23 were more than 28 years old at the time of their first marriage. At α = 0.05, can you support the counselor’s claim? (Adapted from U.S. Census Bureau)

Verified step by step guidance

1. Step 1: Define the null and alternative hypotheses. The null hypothesis (H₀) states that the median age of men at the time of their first marriage is 28 years old (or less). The alternative hypothesis (H₁) states that the median age of men at the time of their first marriage is greater than 28 years old. Mathematically, H₀: Median ≤ 28 and H₁: Median > 28.
2. Step 2: Identify the test to use. Since this is a claim about the median, and the data involves counts of men above and below the median, a non-parametric test such as the sign test is appropriate. The sign test evaluates whether the number of observations above the hypothesized median is significantly greater than expected under the null hypothesis.
3. Step 3: Calculate the test statistic. In the sign test, the test statistic is the number of observations above the hypothesized median. Here, there are 23 men above 28 years old. The total sample size is 56, so the expected number of men above the median under the null hypothesis is 56/2 = 28.
4. Step 4: Determine the critical value or p-value. Use the binomial distribution with n = 56 and p = 0.5 (under the null hypothesis) to calculate the probability of observing 23 or fewer men above the median. Alternatively, you can use a z-approximation for the binomial distribution if the sample size is large enough. The z-score is calculated as z = (x - np) / sqrt(np(1-p)), where x is the observed count, n is the sample size, and p is the probability under the null hypothesis.
5. Step 5: Compare the test statistic to the critical value or p-value. If the p-value is less than the significance level α = 0.05, reject the null hypothesis in favor of the alternative hypothesis. Otherwise, fail to reject the null hypothesis. Conclude whether the counselor's claim is supported based on this comparison.

Key Concepts

Median

The median is a measure of central tendency that represents the middle value in a data set when it is ordered from least to greatest. In the context of this question, the median age of men at first marriage is crucial for determining whether the counselor's claim that it is greater than 28 years old is supported by the sample data.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population based on sample data. In this scenario, the null hypothesis would state that the median age is 28 or less, while the alternative hypothesis posits that it is greater than 28. The outcome of the test will help determine if there is enough evidence to support the counselor's claim.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether to reject the null hypothesis. In this case, α = 0.05 indicates a 5% risk of concluding that a difference exists when there is none. This level helps assess the strength of the evidence against the null hypothesis in the context of the sample data.