Question

Fairfield Homes is developing two parcels near Pigeon Fork, Tennessee. They would like to test whether the mean family income is different at the two parcels. The mean family income for a sample 17 families at the first development is $159,000 with a sample standard deviation of $42,000. A sample of 24 families at the second development had a mean family income of $177,000 with a sample standard deviation of $30,000. The population standard deviations are unknown but assumed to be equal.

(1)	State the decision rule at the .10 significance level for H0: μ1 = μ2; H1: μ1 ≠ μ2. (Negative amounts should be indicated by a minus sign. Round your answers to 3 decimal places.)

Reject H0 if t < ? or t > ?

(2)	Compute the value of the test statistic. (Negative amount should be indicated by a minus sign. Carry at least 3 decimals in your intermediate calculations.  Round your answer to 2 decimal places.)

Value of the test statistic:

(3)	At the .10 significance level, can Fairfield conclude that the population means are different?

Do not reject or Reject H0 ? Fairfield (can or cannot) conclude that the population means are different.

Answer 1

Given that,

For 1st development : n1 =17, x1-bar = $159000 and s1= $42000

For 2nd development : n2 = 24, x2-bar = $177000, s2 = $30000

The population standard deviations are unknown but assumed to be equal.

Therefore, pooled standard deviation is,

The null and alternative hypotheses are,

H0 : μ1 = μ2

H1 : μ1 ≠ μ2

Degrees of freedom = 17 + 24 - 2 = 39

t-critical values at significance level of 0.10 with 39 degrees of freedom are, tcrit = ± 1.685

Decision Rule : Reject H0 if t < -1.685 or t > 1.685

Test statistic is,

The value of test statistic = t = -1.603

Since, test statistic = -1.603 > -1.685, we do not reject H0. Fairfield cannot conclude that the population means are different.