Question

3. The Atlantic Mortgage Insurance Co. markets residential and commercial loans to customers in the maritime region. Recently, the company's board of directors asked whether the company had experienced a difference in the proportion of loan defaults between residential and commercial customers. To prepare an answer to this question, company officials selected a random sample of 200 residential loans and 120 commercial loans. There were 32 defaulted residential loans, and 27 defaulted commercial loans. TOTAL 28 PTS 128 a) Based on the sample information and a significant level of 5%, is there a difference in the proportion of loan defaults between residential and commercial customers? Residential 200 Commercial 2 0 Sample Size default Rate of default Hypotheses (Words and symbols): (2 pts) H H : : ii. Test Statistic pts) iii. Decision Rule 2 pts) iv. Analysis (4 pts) V. Decision Conclusion (2 pts) b) Find the p-value in (a) and interpret its meaning. (2 pts)

Answer 1

For sample 1, we have that the sample size is N1 = 200, the number of favorable cases is X1 = 32, so then the sample proportion is 1 : = 0.16 For sample 2, we have that the sample size is N2 = 120, the number of favorable cases is X2 = 27, so then the sample proportion is f2 = x = 120 = 0.225 The value of the pooled proportion is computed as p = X1 + X2 32+27 N1+N2 = 200-120 = 0.1844 12 Also, the given significance level is a = 0.05. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: P1 = P2 Ha: P1 # P2 This corresponds to a two-tailed test, for which a z-test for two population proportions needs to be conducted (2) Rejection Region Based on the information provided, the significance level is a = 0.05, and the critical value for a two- tailed test is ze = 1.96. The rejection region for this two-tailed test is R= {z:12 > 1.96

(3) Test Statistics The Z-statistic is computed as follows: 2= = Z Ộ1 – Ộ2 VP(1 – )(1/n1 + 1/n2) =- 0.16 – 0.225 0.1844 · (1 – 0.1844)(1/200 + 1/120) = -1.452 (4) Decision about the null hypothesis Since it is observed that z = 1.452 < ze = 1.96, it is then concluded that the null hypothesis is not rejected. Using the P-value approach: The p-value is p= 0.1466, and since p= 0.1466 > 0.05, it is concluded that the null hypothesis is not rejected. (5) Conclusion It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population proportion pı is different than p2, at the 0.05 significance level. Confidence Interval The 95% confidence interval for P1 – P2 is: -0.155 < P1 - P2 < 0.025.