Question

A real estate agency says that the mean home sales price in City A is the same as in City B. The mean home sales price for 30 homes in City A is ​$127,402. Assume the population standard deviation is ​$25,880. The mean home sales price for 30 homes in City B is ​$112, 276. Assume the population standard deviation is ​$27,112. At alpha =0.01​, is there enough evidence to reject the​ agency's claim?

Find the critical​ value(s) and identify the rejection region. Select the correct choice below and fill in the answer box within your choice.

​(Round to three decimal places as​ needed.)

Find the standardized test statistic z.

z=?

​(Round to two decimal places as​ needed.)

Answer 1

Solution:

The null and alternative hypotheses are as follows:

  i.e. The mean home sales price in city A is equal to the mean home sales price in city B.

i.e. The mean home sales price in city A is not equal to the mean home sales price in city B

Since, population standard deviations are known, therefore we shall use z-test for the equality of two population means to test the hypothesis. The test statistic is given as follows:

Where, are sample means, σ_{​​​​​A}^​​​ and σ_{​​​​​B} are population standard deviations and n_{​​​​​​1}, n_{​​​​​​2} are sample sizes.

We have, x̄_{​​​​​A} = ​$127402, σ_{​​​​​A} = $25880, n_{​​​​​​1} = 30

x̄_​​​B = ​$112276, σ_{​​​​B} = $​27112, n_{​​​​​2} = 30

The value of the test statistic is 2.21.

Since, our test is two-tailed test therefore, we shall obtain two-tailed critical value. The two-tailed critical values at α = 0.01 are given as follows:

and  

Critical values are -2.58 and 2.58.

Rejection regions:

Z < Z^*_(0.01/2) and Z > Z^*_{(1- 0.01/2)}

i.e. rejection regions are Z < -2.58 and Z > 2.58.

For two-tailed z-test, atα = 0.01 we make decision rule as follows:

If  Z < Z^*_(0.01/2)   ​​or Z > Z^*_{(1- 0.01/2)}, then we reject the null hypothesis at 0.01 significance level.

If Z^*_(0.01/2) < Z < Z^*_{(1 - 0.01/2)}, then we fail to reject the null hypothesis at 0.01 significance level.

We have, Z = 2.21, which lies between -2.58 and 2.58.

Since, Z^*_(0.01/2) < Z < Z^*_{(1 - 0.01/2)}, therefore we shall be fail to reject the null hypothesis at 0.01 significance level.

Conclusion: At α = 0.01, there is not enough evidence to reject the agency claim that the mean home sales price in City A is the same as in City B.