TABLE 10-4
A real estate company is interested in testing whether, on average, families in Gotham have been living in their current homes for less time than families in Metropolis have. 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 Table 10-4, suppose α = 0.01. 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

TABLE 10-4
A real estate company is interested in testing whether, on average, families in Gotham have been living in their current homes for less time than families in Metropolis have. 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 Table 10-4, suppose α = 0.01. 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 need to conduct a two-sample t-test with equal variances to compare the mean lengths of residence in current homes for families in Gotham and Metropolis. The null hypothesis is that there is no difference in the mean lengths of residence between the two populations, while the alternative hypothesis is that families in Gotham have been living in their current homes for less time than families in Metropolis.

Using the data provided in Table 10-4, we can calculate the pooled sample variance as:

s^2 = [(n1-1)s1^2 + (n2-1)s2^2] / (n1 + n2 - 2)
s^2 = [(100-1)(144.92) + (150-1)(188.59)] / (100+150-2) = 167.71

where s1 and s2 are the sample standard deviations for Gotham and Metropolis, respectively.

The test statistic for the two-sample t-test with equal variances is:

t = (x1 - x2) / [s * sqrt(1/n1 + 1/n2)]

where x1 and x2 are the sample means for Gotham and Metropolis, respectively, and s is the pooled sample standard deviation.

From Table 10-4, we have:

x1 = 5.10, x2 = 6.87, n1 = 100, n2 = 150

Plugging in the values, we get:

t = (5.10 - 6.87) / [sqrt(167.71) * sqrt(1/100 + 1/150)]
t = -4.438

Using a two-tailed t-test at the 0.01 level of significance with 248 degrees of freedom (100 + 150 - 2), the critical value is ±2.612. Since our calculated t-value of -4.438 is outside this range, we reject the null hypothesis and conclude that there is enough evidence to support that the mean amount of time families in Gotham have been living in their current homes is less than families in Metropolis. Thus, the correct answer is B).

TABLE 10-4 a Real Estate Company Is Interested in Testing Whether, on Whether