Question

I 33. Under water. In early 2012, the proportion of mortgages that were “under water"-a negative equity position in which the homeowner owes more than the value of the home-was highest in Nevada and Arizona (s.wsj.net/public/ resources/documents/info-NEGATIVE_EQUITY_0911 .html). Lenders are also interested in homes at risk-those within 5% of being in a negative equity position. A sample of mortgages from these two states suggest that the prob- lem may be less severe in Nevada. The sample found that 62 of 1369 Arizona mortgages were in a "near negative equity" state. A sample of 604 Nevada mortgages found 22 in a near negative equity state. a) Is there evidence that the percentage of near negative equity mortgages is different in the two states? b) Give a 90% confidence interval for the difference in proportions. 34. Labor force. Immigration reform has focused on dividing illegal immigrants into two groups: long-term and short-term. In a random sample of 958 construction workers from the Northeast, 66 are illegal short-term immigrants. In the Midwest, 42 out of a sample of 1070 are illegal short-term immigrants. a) Is there evidence that the percentage of construction workers who are illegal short-term immigrants differs in the two regions? b) Give a 90% confidence interval for the difference in proportions.

Answer 1

33.
a)

Let p1 and p2 be the percentage of near negative equity mortgages in Arizona and Nevada respectively.
Null hypothesis H0: p1 - p2 = 0
Alternative hypothesis Ha: p1 - p2 $\ne$ 0

For Arizona, x1 (success) = 62

failures, n1 - x1 = 1369 - 62 = 1301

For Nevada, x2 (success) = 22

failures, n2 - x2 = 604 - 22 = 582

Since each sample includes at least 10 successes and 10 failures, we can use two-proportion z-test.

For this analysis, the significance level is 0.1 (90% confidence interval).

Pooled proportion, p = (x1 + x2) / (n₁ + n₂)

= (62 + 22) / (1369 + 604) = 0.04257476

Standard error of pooled proportion, SE = sqrt{ p * ( 1 - p ) * [ (1/n₁) + (1/n₂) ] }

SE = sqrt{ 0.04257476 * ( 1 - 0.04257476 ) * [ (1/1369) + (1/604) ] } = 0.009862156

p1 = x1 / n1 = 62 / 1369 = 0.04528853

p2 = x2 / n2 = 22 / 604 = 0.03642384

Test statistic, Z = (p1 - p2) / SE = (0.04528853 - 0.03642384) / 0.009862156 = 0.8989

For two tail test, p-value = 2 * P(z > 0.8989) = 0.3687

Since, p-value is greater than 0.1 significance level, we fail to reject null hypothesis H0 and conclude that there is no strong evidence that percentage of near negative equity mortgages is different in two states.

b)

z value for 90% confidence interval is 1.28

Margin of error = z * SE = 1.28 * 0.009862156 = 0.0126

Point estimate of the percentage difference = p1 - p2 = 0.04528853 - 0.03642384 = 0.0089

90% confidence interval of the difference in percentage of near negative equity mortgages in two states is,

(0.0089 - 0.0126, 0.0089 + 0.0126)

(-0.0037,  0.0215)

34.

a)

Let p1 and p2 be the percentage of construction workers who are illegal short-term migrants in two regions respectively.
Null hypothesis H0: p1 - p2 = 0
Alternative hypothesis Ha: p1 - p2 $\ne$ 0

For NorthEast, x1 (success) = 66

failures, n1 - x1 = 958 - 66 = 892

For MidWest, x2 (success) = 42

failures, n2 - x2 = 1070 - 42 = 1028

Since each sample includes at least 10 successes and 10 failures, we can use two-proportion z-test.

For this analysis, the significance level is 0.1 (90% confidence interval).

Pooled proportion, p = (x1 + x2) / (n₁ + n₂)

= (66 + 42) / (958 + 1070) = 0.05325444

Standard error of pooled proportion, SE = sqrt{ p * ( 1 - p ) * [ (1/n₁) + (1/n₂) ] }

SE = sqrt{ 0.05325444 * ( 1 - 0.05325444 ) * [ (1/958) + (1/1070) ] } = 0.009987433

p1 = x1 / n1 = 66 / 958 = 0.06889353

p2 = x2 / n2 = 42 / 1070 = 0.03925234

Test statistic, Z = (p1 - p2) / SE = (0.06889353 - 0.03925234) / 0.009987433 = 2.9678

For two tail test, p-value = 2 * P(z > 2.9678) = 0.003

Since, p-value is less than 0.1 significance level, we reject null hypothesis H0 and conclude that there is significantly strong evidence that percentage of construction workers who are illegal short-term migrants in two regions are different.

b)

z value for 90% confidence interval is 1.28

Margin of error = z * SE = 1.28 * 0.009987433 = 0.0128

Point estimate of the percentage difference = p1 - p2 = 0.06889353 - 0.03925234 = 0.0296

90% confidence interval of the difference in percentage of near negative equity mortgages in two states is,

(0.0296 - 0.0128, 0.0296 + 0.0128)

(0.0168,  0.0424)