Question

The paper "Ready or Not? Criteria for Marriage Readiness among Emerging Adults" (Journal of \(\underline{\text { Ado- }}\) lescent Research [2009]: 349-375) surveyed emerging adults (defined as age 18 to 25 ) from five different colleges in the United States. Several questions on the survey were used to construct a scale designed to measure endorsement of cohabitation. The paper states that "on average, emerging adult men \((\mathrm{M}=3.75, \mathrm{SD}=1.21)\) reported higher levels of cohabitation endorsement than emerging adult women \((\mathrm{M}=3.39, \mathrm{SD}=1.17) . "\) The sample sizes were 481 for women and 307 for men. a. Carry out a hypothesis test to determine if the reported difference in sample means provides convincing evidence that the mean cohabitation endorsement for emerging adult women is significantly less than the mean for emerging adult men for students at these five colleges. b. What additional information would you want in order to determine whether it is reasonable to generalize the conclusion of the hypothesis test from Part (a) to all college students?

Short answer

The outcome of the hypothesis test will determine if there is evidence that the mean cohabitation endorsement for emerging adult women is significantly less than the mean for emerging adult men. To generalize the conclusion to all college students, additional information such as sampling methods, non-response biases, and college diversity would be required.

Step-by-step solution

State the Hypotheses

The null hypothesis \(H_0\) is that the mean cohabitation endorsement of emerging adult men and women are equal. The alternate hypothesis \(H_a\) is that the mean cohabitation endorsement of emerging adult women is less than emerging adult men:\n\n\( H_0 : \mu_{men} = \mu_{women} \)\n\n\( H_a : \mu_{men} > \mu_{women} \)

Compute Test Statistic

The test statistic for a two-sample t-test can be calculated by the following formula: \n\n\[ t = \frac{{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}}{{\sqrt{(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2})}}} \]\n\nSubstitute the given values into the formula: \n\n\[ t = \frac{{(3.39 - 3.75) - 0}}{{\sqrt{(\frac{1.17^2}{481} + \frac{1.21^2}{307})}}} \]

Determine the p-value

Use the calculated t statistic to find the p-value from a t-distribution table. The degrees of freedom would be the smaller of \( n_1 - 1 \) and \( n_2 - 1 \), which is \( 307 - 1 = 306 \). If p-value is less than 0.05 (5% significance level), reject the null hypothesis.

Conclude the Hypothesis Test

If the p-value is less than the significance level (0.05), would reject the null hypothesis and conclude that there is convincing evidence that the mean cohabitation endorsement for emerging adult women is significantly less than the mean for emerging adult men among the students at these five colleges.

Additional Information

In order to generalize the conclusion to all college students, information regarding the selection of students - specifically, if the sample selected was random without biases or if it is representative of all college students of emerging adult age - would be needed. Additionally, information about the survey's response rate and whether there's a potential non-response bias could be valuable. Finally, understanding if the five colleges involved in the study represent the overall diversity of all colleges is essential.

Key concepts

Cohabitation Endorsement

In the context of the study, cohabitation endorsement refers to the extent to which individuals agree with or support the idea of living together without being married. This concept can vary widely, influenced by cultural, educational, and personal beliefs.
The survey in question used a series of questions to gauge cohabitation endorsement among emerging adults. It’s important to understand how endorsement levels might impact societal norms and relationship dynamics among this age group.
Understanding cohabitation endorsement is vital, as it reflects broader social trends and can affect attitudes towards marriage and traditional family structures. As societal views on marriage evolve, the endorsement may vary significantly among different demographic groups, including gender differences as observed in the study.

Emerging Adults

The term emerging adults refers to individuals typically aged 18 to 25 who are transitioning from adolescence to adulthood. This period is characterized by exploration and self-discovery.
According to developmental psychologists, emerging adults are exploring their identities, career paths, and relationships. This unique stage of life can lead to distinctive attitudes and behaviors, such as differences in cohabitation endorsement.
The study highlighted that emerging adults at several U.S. colleges perceive cohabitation differently, which may stem from their ongoing identity exploration and the changing societal norms around this age group. Understanding their outlook can provide insights into future societal trends and potential shifts in marriage and family patterns.

Sample Means Comparison

Sample means comparison involves statistically comparing the average (mean) results of different groups to determine if there are significant differences. This is often done using hypothesis testing.
In the study, the hypothesis test aimed to compare the mean cohabitation endorsements between men and women. A t-test was used to see if emerging adult men had a higher endorsement score than women.
The formula for the t-statistic compares the means and considers the variances and sample sizes of both groups. This approach helps determine if the observed differences are statistically significant or merely due to random chance, enabling us to draw conclusions about the population from which the sample was drawn.

P-Value Interpretation

The p-value is a crucial component in hypothesis testing, representing the probability that the observed data would occur if the null hypothesis is true.
In the context of the study, the p-value helps determine whether the difference in cohabitation endorsement between men and women is statistically significant. A low p-value (typically less than 0.05) indicates that the observed difference is unlikely due to chance alone, leading to the rejection of the null hypothesis.
Interpreting the p-value correctly is essential for making informed decisions based on the data. In this study, if the p-value is below the threshold, it suggests strong evidence against the null hypothesis, supporting the conclusion that men and women differ significantly in their levels of cohabitation endorsement.