Question

If a random sample of 20 homes south of a town has a mean selling price of $145,225 and a standard deviation of $4600, and a random sample of 20 homes north of a town has a mean selling price of $148,575 and a standard deviation of $5700, can you conclude that there is a significant difference between the selling price of homes in these two areas of the town at the 0.05 level? Assume normality.

(a) Find t. (Round your answer to two decimal places.)


(ii) Find the p-value. (Round your answer to four decimal places.)


(b) State the appropriate conclusion.

Fail to reject the null hypothesis. There is not significant evidence of a difference in means. Reject the null hypothesis. There is significant evidence of a difference in means.     Fail to reject the null hypothesis. There is significant evidence of a difference in means. Reject the null hypothesis. There is not significant evidence of a difference in means.

Answer 1

The provided sample means are shown below:

Also, the provided sample standard deviations are:

  

and the sample sizes are n1 ​= 20 and n2 = 20

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ1​ = μ2​

Ha: μ1​ ≠ μ2​

This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

(2) Rejection Region

Based on the information provided, the significance level is α = 0.05, and the degrees of freedom are df = 38. In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal:

Hence, it is found that the critical value for this two-tailed test is t_c = 2.024 for α = 0.05 and df = 38.

(3) Test Statistics

Since it is assumed that the population variances are equal, the t-statistic is computed as follows:

t =−2.045

(4) Decision about the null hypothesis

Since it is observed that |t| = 2.045 > t_c = 2.024, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p = 0.0478, and since p = 0.0478 < 0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean μ1​ is different than μ2​, at the 0.05 significance level.

(a) Find t. (Round your answer to two decimal places.)
t =−2.04

(ii) Find the p-value. (Round your answer to four decimal places.)

p = 0.0478
(b) State the appropriate conclusion.

Reject the null hypothesis. There is significant evidence of a difference in means.