SCENARIO 10-3
A real estate company is interested in testing whether the mean time that families in Gotham have been living in their current homes is less than families in Metropolis.Assume that the two population variances are equal.A random sample of 100 families from Gotham and a random sample of 150 families in Metropolis yield the following data on length of residence in current homes. Gotham: x̄ G = 35 months,S G 2 = 900 Metropolis: x̄ M = 50 months,S M 2 = 1050
-Referring to Scenario 10-3,what is the test statistic for the difference between sample means?
A)- 8.75
B)- 3.69
C)- 2.33
D)- 1.96

Question

SCENARIO 10-3
A real estate company is interested in testing whether the mean time that families in Gotham have been living in their current homes is less than families in Metropolis.Assume that the two population variances are equal.A random sample of 100 families from Gotham and a random sample of 150 families in Metropolis yield the following data on length of residence in current homes.  Gotham: x̄ G = 35 months,S G 2 = 900 Metropolis: x̄ M = 50 months,S M 2 = 1050
-Referring to Scenario 10-3,what is the test statistic for the difference between sample means?
A)- 8.75
B)- 3.69
C)- 2.33
D)- 1.96

Quizplus · Accepted Answer

Since the population variances are assumed to be equal, we can use the pooled variance estimator: 
Sp² = ((n1 - 1)S1² + (n2 - 1)S2²) / (n1 + n2 - 2) 
Plugging in our values, we get: 
Sp² = ((100 - 1)900 + (150 - 1)1050) / (100 + 150 - 2) = 980.39 
Next, we calculate the standard error of the difference between sample means: 
SE = sqrt(Sp² * (1/n1 + 1/n2)) = sqrt(980.39 * (1/100 + 1/150)) = 8.625 
Finally, we calculate the test statistic: 
t = (x̄1 - x̄2) / SE = (35 - 50) / 8.625 = -1.739 
Using a two-tailed test at the 5% significance level with degrees of freedom of 100 + 150 - 2 = 248, the critical value of t is approximately ±1.97. Because our calculated test statistic (-1.739) does not exceed the critical value, we fail to reject the null hypothesis.

SCENARIO 10-3 a Real Estate Company Is Interested in Testing Whether the Whether