Question

According to a new report from the Wall Street Journal, female real estate agents sell more homes per year than male real estate agents do. You, the head of a large real estate company, want to know if this is true within your organization, so you sample 55 women and 37 men. From the sample, you find that the female agents sell an average of 25.2 homes per year with a standard deviation of 7.7 homes. The male agents sell an average of 22.9 homes per year with a standard deviation of 6.3 homes. Let females be population 1 and males be population 2 and assume the population variances are equal. Use a significance level of 0.05. Find and enter the correct critical value needed to test your claim at the 5% significance level. Round your final answer to 4 decimals. Hint: Things you will need to know: alpha, df, is this one or two-tailed test, and which table to use!

Answer 1

Answer 2

Ans.

The given situation can be studied under the Two sample t-test with EQUAL VARIANCE and a one tailed test. As its only asking that whether the avg home sell for female agent is greater than that of male agent or not, that's why its a one tailed test.

We have to find that the average home sell per year for female agent is greater than the male agent or not. As the information given in the question with two samples one for female and one for male with their corresponding mean and standard deviation.

since it is the case of two sample t-test with Equal variance, from data given in question:

For female sample:

According to a new report from the Wall Street Journal, female real estate agents sell more homes per year than male real estate agents do. You, the head of a large real estate company, want to know if this is true within your organization, so you sample 55 women and 37 men. From the sample, you find that the female agents sell an average of 25.2 homes per year with a standard deviation of 7.7 homes. The male agents sell an average of 22.9 homes per year with a standard deviation of 6.3 homes. Let females be population 1 and males be population 2 and assume the population variances are equal. Use a significance level of 0.05. Find and enter the correct critical value needed to test your claim at the 5% significance level. Round your final answer to 4 decimals. Hint: Things you will need to know: alpha, df, is this one or two-tailed test, and which table to use!

, avg sell of home for female agent per year for given sample.

For male sample:

,   avg sell of home for male agent per year for given sample.

Hypothesis:

,   for population the difference in avg sell of home per year for female and male is zero.

,      for population the avg sell of home per year by female agent is greater than that of male.

avg sell of home per by female agent at population level.

avg sell of home per by male agent at population level.

Formula for t-statistic:    

$t=rac{ar{y}_{1}-ar{y}_{2}}{sqrt{rac{s_{p}^{2}}{n_{1}}+rac{s_{p}^{2}}{n_{2}}}}$

         ,       $s_{p}^{2}:$ known as the pooled variance.

$s_{p}^{2}=rac{3201.66+1428.84}{54+36}$

$s_{p}^{2}=rac{4630.5}{90}$

$s_{p}^{2}=51.45$

Now using the value $s_{p}^{2}=51.426$ , in the t-test formula we now calculate the t-statistic for the two sample t-test.

$t=rac{ar{y}_{1}-ar{y}_{2}}{sqrt{rac{s_{p}^{2}}{n_{1}}+rac{s_{p}^{2}}{n_{2}}}}$

$t=rac{ar{y}_{1}-ar{y}_{2}}{sqrt{rac{s_{p}^{2}}{n_{1}}+rac{s_{p}^{2}}{n_{2}}}}=rac{25.2-22.9}{sqrt{rac{51.45}{55}+rac{51.45}{37}}}$

t-1.508076N 1.5080

Now we have to find the $t_{critical}$ value for significance level of , before that we need to find the degfree of freedom(df).

a-0.05

$df=n_{1}+n_{2}-2$

$df=55+37-2=90$

Using a t table we can calculate the $t_{critical}$ value for one tailed test(right tail ), with df=90, .

a-0.05

So we have our $mathbf{t_{critical}=1.662},$

tcalculated-1.5080

${t_{calculated}<t_{critical}$

Since the ${t_{calculated}<t_{critical}$ , we are FTR(Fail TO Reject) Null hypothesis $H_{0}$ i.e., , and can conclude that the average sell of home for female agent population is equal to the average sell of home for male agent population per year.