Question

Many students graduate from college deeply in debt from student loans, credit cards, and so on. Curious about the impact of debt on decisions about the future, a WCU sociologist took a random sample of 401 single students at WCU, classified them by a binary view of gender, and asked, “Would you consider marrying someone who was $25,000 or more in debt?” The results of this survey are shown in the following table. Test at a 1% significance level if men and women responded differently from one another. In other words, are gender and response related?

WOMEN, YES = 125

WOMEN, NO = 59

WOMEN, UNCERTAIN = 21

MEN, YES = 101

MEN, NO = 79

MEN, UNCERTAIN = 16

If This Problem Calls for:	Use Test Statistic	Use Degrees of Freedom
Pooled Variance Test	3.765	Calculate your own
Separate Variance Test	3.415	df > 75
Paired Samples Test	5.888	Calculate your own
Chi-squared Test	Calculate your own	Calculate your own

Determine which type of test is required for each question and clearly state the type of test that is called for. Then, perform all steps of the test. Show the critical value approach and the p-value approach. Explain your conclusions at the end of each test.

Answer 1

Answer:

The Chi-square test of independence is used.

Chi-square statistic = 5.9239

Degree of freedom = 2

Explanation:

Since we are comparing two categorical variables such that gender and the marriage decision, the Chi-square test of independence is used to test the following hypothesis,

Hypothesis:

Null hypothesis: The two variables are independent

Alternative hypothesis: The two variables are dependent

Critical value

The critical value for the chi-square statistic is obtained from the chi-square critical value table for the degree of freedom = (r-1)(c-1) = (3-1)(2-1)=2 for significance level = 0.01, the critical value is,

Test statistic

The observed values are,

.	Women	Men	Total
Yes	125	101	226
No	59	79	138
Uncertain	21	16	37
Total	205	196	401

The expected values are obtained using the formula,

.	Women	Men	Total
Yes	226*205/401=115.5362	226*196/401=110.4638	226
No	138*205/401=70.5486	138*196/401=67.4514	138
Uncertain	37*205/401=18.9152	37*196/401=18.0848	37
Total	205	196	401

The Chi-Square value is obtained using the formula

Observed, Oi	Expected, Ei	(Oi-Ei)^2/Ei
125	115.5362	0.7752
59	70.5486	1.8905
21	18.9152	0.2298
101	110.4638	0.8108
79	67.4514	1.9773
16	18.0848	0.2403
.	Sum	5.9239

P-value:

The p-value is obtained from the chi-square distribution table for chi-square = 5.9239 and degree of freedom = 2

P-value = 0.0517

Decision

Critical value approach

Since the chi-square value is less than the critical value, the null hypothesis is not rejected at a 1% significance level.

P-value approach

Since the P-value is greater than 0.01 at a 1% significance level, the null hypothesis is not rejected.

Conclusion:

There is not sufficient evidence to conclude that the variables gender and marriage decision are dependent.