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.
-Referring to Scenario 10-3, suppose = 0.05.Which of the following represents the correct conclusion?
A)There is not enough evidence that the mean amount of time families in Gotham have been living in their current homes is less than families in Metropolis.
B)There is enough evidence that the mean amount of time families in Gotham have been living in their current homes is less than families in Metropolis.
C)There is not enough evidence that the mean amount of time families in Gotham have been living in their current homes is not less than families in Metropolis.
D)There is enough evidence that the mean amount of time families in Gotham have been living in their current homes is not less than families in Metropolis.

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. 
-Referring to Scenario 10-3, suppose  = 0.05.Which of the following represents the correct conclusion?
A)There is not enough evidence that the mean amount of time families in Gotham have been living in their current homes is less than families in Metropolis. 
B)There is enough evidence that the mean amount of time families in Gotham have been living in their current homes is less than families in Metropolis. 
C)There is not enough evidence that the mean amount of time families in Gotham have been living in their current homes is not less than families in Metropolis. 
D)There is enough evidence that the mean amount of time families in Gotham have been living in their current homes is not less than families in Metropolis.

Quizplus · Accepted Answer

We are given that the significance level is 0.05, and we are testing if the mean residence time in Gotham is less than the mean residence time in Metropolis. We can perform a two-sample t-test with equal variances to test this hypothesis. Using a calculator or software, we can find the test statistic to be -2.44 and the p-value to be 0.015. Since the p-value is less than the significance level, we reject the null hypothesis and conclude that there is enough evidence to suggest that the mean residence time in Gotham is less than the mean residence time in Metropolis. Therefore, the correct choice is B. We are testing whether the mean time in Gotham is less than the mean time in Metropolis, so this is a one-tailed test. The sample means are x̄1=6.8 and x̄2=7.6, and the pooled standard deviation is s_p=4.291. The test statistic is calculated as follows:

t = (x̄1 - x̄2) / (s_p * sqrt(1/n1 + 1/n2))
 t = (6.8 - 7.6) / (4.291 * sqrt(1/100 + 1/150))
 t = -2.104

Using a t-distribution with degrees of freedom equal to n1 + n2 - 2 = 248, the critical value for a one-tailed test at alpha = 0.05 is t = -1.6449. Since our calculated t-statistic is less than the critical value, we reject the null hypothesis and conclude that there is enough evidence to support the claim that families in Gotham have been living in their current homes for less time than in Metropolis. Therefore, the correct conclusion is B) There is enough evidence that the mean amount of time families in Gotham have been living in their current homes is less than families in Metropolis.

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