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 ​$127,466. Assume the population standard deviation is ​$25,877. The mean home sales price for 30 homes in City B is ​$112,349. Assume the population standard deviation is ​$27,108. At alpha=0.01​, is there enough evidence to reject the​ agency's claim? Complete parts​ (a) through​ (d) below.
(a) Identify the claim and state H_0 and_1Ha. What is the claim?

A. The mean home sales price in City A is greater than as in City B.
B. The mean home sales price in City A is less than in City B.
C. The mean home sales price in City A is the same as in City B.
D. The mean home sales price in City A is not the same as in City B.

Answer 1

Answer:

a) The mean home sales price in City A is the same as in City B

The calculated value t = 2.1731 < 2.3924 at 0.01 level of significance

The null hypothesis is accepted

Step-by-step explanation:

Step(i):-

Given data The mean home sales price for 30 homes in City A is ​$127,466

Mean of the first sample(x₁⁻) = $127,466

Standard deviation of the first sample(s₁) =  ​$25,877

The mean of the second sample(x₂⁻) = ​$112,349

standard deviation of the second sample(s₂) = ​$27,108

sample size n₁ = n₂ = 30

Step(ii):-

Null hypothesis : H₀ : The mean home sales price in City A is the same as in City B.

Alternative Hypothesis:H₁: The mean home sales price in City A is the not same as in City B.

The test statistic

$Z = \frac{x^{-} _{1}- x^{-} _{2} }{\sqrt{S^{2} (\frac{1}{n_{1} }+\frac{1}{n_{2} } } }$

where

            $S^{2} = \frac{n_{1}S_{1} ^{2} + n_{2} S_{2} ^{2} }{n_{1} +n_{2}-2 }$

           $S^{2} = \frac{30 X (25,877)^{2} + 30 X (27,108)^{2} }{30+30-2 }$

On calculation ,we get

          $s^{2} = 726,446,272.24$

Test statistic

     $t = \frac{127,466 -112,349 }{\sqrt{726,446,272.24 (\frac{1}{30}+\frac{1}{30 } }) }$

     t = 2.1731

Degrees of freedom

                            γ = n₁ +n₂ -2

                             γ  = 30+30-2 =58

$Z_{\frac{\alpha }{2} } = Z_{\frac{0.01}{2} } = Z_{0.005} = 2.3924$

Conclusion:-

The calculated value t = 2.1731 < 2.3924 at 0.01 level of significance

The null hypothesis is accepted

The mean home sales price in City A is the same as in City B.