Question

Toll Brothers is a luxury home builder that would like to test the hypothesis that the average size of new homes exceeds 2,400 square feet. A random sample of 36 newly constructed homes had an average of 2,510 square feet with a sample standard deviation of 480 square feet. Conduct a test of hypothesis at a 1% level of significance to see if indeed the average size of new homes exceeds 2,400 square feet.

Answer 1

Answer:

Since the pvalue of the test is 0.09 > 0.01, we do not reject the null hypothesis that the average size of new homes is of 2400 square feet.

Step-by-step explanation:

Toll Brothers is a luxury home builder that would like to test the hypothesis that the average size of new homes exceeds 2,400 square feet. Test to see if the average size of new homes exceeds 2,400 square feet.

At the null hypothesis, we test if the average is of 2400 square feet, that is:

$H_o: \mu = 2400$

At the alternate hypothesis, we test if the average is greater than 2400 square feet, that is:

$H_a: \mu > 2400$

The test statistic is:

Since we have the standard deviation for the sample, we use the t-distribution.

$t = \frac{X - \mu}{\frac{s}{\sqrt{n}}}$

In which X is the sample mean, $\mu$ is the value tested at the null hypothesis, s is the standard deviation of the sample and n is the size of the sample.

2400 is tested at the null hypothesis:

This means that $\mu = 2400$

A random sample of 36 newly constructed homes had an average of 2,510 square feet with a sample standard deviation of 480 square feet.

This means that $n = 36, X = 2510, s = 480$

Test statistic:

$t = \frac{X - \mu}{\frac{s}{\sqrt{n}}}$

$t = \frac{2510 - 2400}{\frac{480}{\sqrt{36}}}$

$t = 1.375$

Pvalue of the test and decision:

The pvalue of the test is the probability of finding a sample mean of at least 2510, which is the pvalue of t = 1.375 with 36 - 1 = 35 degrees of freedom, using a one-tailed test.

With the help of a calculator, this pvalue is of 0.09

Since the pvalue of the test is 0.09 > 0.01, we do not reject the null hypothesis that the average size of new homes is of 2400 square feet.