Question

A real estate agent claims that there is no difference between the mean household incomes of two
neighborhoods. The mean income of 12 randomly selected households from the first neighborhood was
$32,750 with a standard deviation of $1900. In the second neighborhood, 10 randomly selected households
had a mean income of $31,200 with a standard deviation of $1825. At , does the realtor have a valid
claim? Assume population variances are not equal.

Answer 1

Assuming the level of significance to be 5%

The provided sample means are shown below:

Also, the provided sample standard deviations are:

and the sample sizes are

.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: ​

Ha: ​

This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df=19.551. In fact, the degrees of freedom are computed as follows, assuming that the population variances are unequal.

Hence, it is found that the critical value for this two-tailed test is tc​=2.089, for α=0.05 and df=19.551.

The rejection region for this two-tailed test is R={t:∣t∣>2.089}.

(3) Test Statistics

Since it is assumed that the population variances are unequal, the t-statistic is computed as follows:

(4) The decision about the null hypothesis

Since it is observed that ∣t∣=1.947≤tc​=2.089, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.0661, and since p=0.0661≥0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ1​ is different than μ2​, at the 0.05 significance level.

Graphically

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