Question

A real estate agent claims that the mean living area of all single-family homes in their area is at most 2400 square feet. A random sample of 41 homes in the area showed a mean living area of 2540 square feet with a standard deviation of 472 square feet.

a) Using a 1% level of significance, perform a complete hypothesis test to determine if the real estate agent’s claim is true. Use the critical value method.

b) What would a Type I error be in this case? What is the probability of making this error?

c) Will your conclusion of part a) change if the probability of making a Type I error is zero? Explain.

Answer 1

Let X be the event of living are of all single families

Summary of sample of 41 families

The test is to check whether the living area is at most 2400, that is, lesser than or equal to 2400.

Therefore this is a 1-tailed t-test for population mean since we don't have population std deviation and we're just asked to check if it is less or not.

Where is the population mean and

There is '=' in null hypothesis since it always should have equality sign and '>' is included since null hypo is the complement of the original claim (alternative hypo)

(a)

Test Statistic:

Critical value at 1% level of significance

Decision Criteria: Reject if T.S. > C. V.

1.8992 < 2.7045

Decision: Do not reject at 1% level of significance.

Conclusion: The living area of single families is more than 2400 sq ft.

(b)

Type 1 error is rejecting null hypothesis when it is true. In our case

We reject the claim the mean living area of all single families is 2400 even though it is true.

is the probability of committing type 1 error.  To calculate we look up for value of Test Stat in the t-dist table

=

= 0.0324

The probability of committing a type 1 error is 0.0324.

(c)

= 0 tells us that there 0% chance of reject the null hypothesis when it is true. That is, 0% of rejecting the decision .when null hypothesis is true.

Since our hypothesis decision is to not reject the null hypothesis, an = 0 will therefore not change our conclusion.