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The paper "Ready or Not? Criteria for Marriage Readiness among Emerging Adults" (journal of Adolescent 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 \((M=3.75, S D=1.21)\) reported higher levels of cohabitation cndorsement than cmerging 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?

Step by step solution

01

State the Hypotheses

The null hypothesis \(H_0\) assumes that there is no difference between the population means of men and women. So, \(H_0: \mu_m = \mu_w\). The alternative hypothesis \(H_1\) is that the population mean for women is less than that for men, \(H_1: \mu_w < \mu_m\).

02

Calculate Difference in Sample Means

Calculate the difference between sample means: \(d = M_w - M_m = 3.39 - 3.75 = -0.36.\)

03

Calculate Standard Error

The formula for standard error for two means is \(SE = \sqrt{ \frac{{SD_w^2}}{{n_w}} + \frac{{SD_m^2}}{{n_m}} }\). Plugging the given values, \(SE = \sqrt{ \frac{{1.17^2}}{{481}} + \frac{{1.21^2}}{{307}} }.\)

04

Calculate t-statistic

Here, t-statistic \(t = \frac{d}{SE}\). Calculate this value using the earlier determined values of \(d\) and \(SE\). This gives the number of standard errors the difference in sample means is away from the null hypothesis mean difference of 0.

05

Find the P-value

Looking up the calculated t-statistic in a t-table (or using a t-distribution calculator), find the corresponding P-value. This P-value gives the probability under the null hypothesis of getting a t-statistic as extreme as the one calculated. If this is a small value (traditionally less than 0.05), we have evidence to reject the null hypothesis.

06

State Conclusion

If P-value is less than 0.05, we reject the null hypothesis, and conclude that there is significant evidence to state that the mean endorsement for cohabitation among women is lower than that of men. If P-value is greater than or equal to 0.05, we fail to reject the null hypothesis, and conclude that there is not significant evidence to prove that mean endorsement for cohabitation among women is lower than that of men.

07

Additional Information for Generalization

For part (b), to apply these results to all college students, it would be helpful to know if the students surveyed were a random sample, how the colleges were selected, the diversity of the students and the colleges in terms of location, socio-economic status, religion, etc., as these factors could affect attitudes towards cohabitation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis ( H₀) and the alternative hypothesis ( H₁) form the foundation for statistical analysis. The null hypothesis represents a position of no effect or no difference, and is the claim we aim to challenge with our data. For instance, in the exercise regarding cohabitation endorsement levels, the null hypothesis asserts that there is no difference in average endorsement levels ( M_w) = M_m) ) between emerging adult men and women.

Conversely, the alternative hypothesis posits that there is an effect or a difference, which has not occurred due to random chance. In the cohabitation study, the alternative hypothesis suggests that emerging adult women have a significantly lower mean endorsement for cohabitation ( M_w) < M_m) ). Establishing hypotheses is crucial as they determine the direction and nature of the statistical tests we will perform.

T-Statistic

The t-statistic is a ratio which allows us to assess how far apart the sample means are, relative to the variation in the sample data. In simpler terms, it helps to determine whether a statistical difference between two groups’ means is likely to have occurred by chance. The t-statistic is calculated using the difference between sample means divided by the standard error. In our example related to cohabitation endorsement, the t-statistic quantifies the number of standard errors the observed difference in means ( d) ) is from the expected difference under the null hypothesis (typically zero). The further away the t-statistic is from zero, the more likely it is that the observed difference did not occur by chance, thus providing evidence against the null hypothesis.

P-Value

In the context of hypothesis testing, the p-value is a probability that measures the strength of the evidence against the null hypothesis provided by the sample data. It represents the probability of obtaining a test statistic as extreme as the one computed, assuming the null hypothesis is true. A low p-value ( p<0.05) ) typically indicates strong evidence against the null hypothesis, suggesting we should reject it in favor of the alternative hypothesis. In our case study, calculating the p-value helps to decide whether the observed difference in cohabitation endorsement between genders is statistically significant or could be attributed to random sampling variation.

Sample Means Comparison

Comparing sample means is a fundamental aspect of hypothesis testing when dealing with quantitative data. It involves assessing whether the mean values observed in two different samples are statistically significantly different from one another. This process involves calculating the difference between the sample means and evaluating this result against a theoretical distribution (usually a t-distribution for small samples). In the exercise, comparing the sample means of cohabitation endorsements between men and women is the key step for testing the hypothesis that there is a gender difference in attitudes towards cohabitation among college students.

Standard Error Calculation

The standard error is a measurement of the variability of a sample mean estimate. It is important in hypothesis testing because it is used in calculating the test statistic, which in turn helps to inform us about the precision of our sample mean difference estimate. To calculate the standard error for comparing two means, as seen in the cohabitation exercise, we use the formula involving the standard deviations of both groups and their respective sample sizes. A smaller standard error suggests that the sample mean is a more reliable estimate of the population mean. Knowing the standard error is essential when assessing the reliability of sample comparisons and in the calculation of the t-statistic.

Statistical Significance

Statistical significance is a term used to decide if the results of the statistical test are likely to be due to something other than mere random chance. It indicates whether the observed effects or differences in the study are likely to be genuine or just due to variability in the data. Significance is often denoted by the p-value; if the p-value is less than the chosen significance level ( < 0.05)) , the results are deemed statistically significant. This is integral to our decision-making in hypothesis testing, such as determining if the difference in cohabitation endorsement by gender in the study really exists or is statistically significant.

Generalization of Results

The ability to generalize results from a study to a larger population is an essential aspect of research. It depends on how the sample was chosen and whether it is representative of the target population. When our hypothesis test indicates a significant result, we must consider whether our sample is random and diverse enough to allow for broader generalization. In the case of the cohabitation endorsement study, knowing the demographic and geographic diversity of the student sample, as well as the selection process of the colleges, is necessary to understand whether these findings can be generalized to all college students. It is only with representative samples that we can reasonably infer findings to a wider population.