In Exercises 17 and 18, (c) find the standardized test statistic t, Assume the samples are random and independent, and the populations are normally distributed.


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 is \$52,750 with a standard deviation of \$2900. In the second neighborhood, 10 randomly selected households have a mean income of \$51,200 with a standard deviation of \$2225. At α=0.01, can you reject the real estate agent’s claim? Assume the population variances are equal.

Verified step by step guidance

1. Identify the hypotheses for the test: the null hypothesis \(H_0\) states that the mean incomes are equal, i.e., \(\mu_1 = \mu_2\), and the alternative hypothesis \(H_a\) states that the means are different, i.e., \(\mu_1 \neq \mu_2\).
2. Since the population variances are assumed equal, calculate the pooled standard deviation \(s_p\) using the formula: \[s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}\] where \(n_1\) and \(n_2\) are the sample sizes, and \(s_1\) and \(s_2\) are the sample standard deviations.
3. Calculate the standard error of the difference between the two sample means using the pooled standard deviation: \[SE = s_p \times \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}\]
4. Compute the test statistic \(t\) using the formula: \[t = \frac{\bar{x}_1 - \bar{x}_2}{SE}\] where \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means.
5. Determine the degrees of freedom for the test as \(df = n_1 + n_2 - 2\), then compare the calculated \(t\) value to the critical \(t\) value from the \(t\)-distribution table at \(\alpha = 0.01\) for a two-tailed test to decide whether to reject the null hypothesis.

Key Concepts

Two-Sample t-Test for Means

This test compares the means of two independent samples to determine if there is a statistically significant difference between their population means. It is appropriate when samples are random, independent, and populations are normally distributed. The test statistic follows a t-distribution under the null hypothesis.

Pooled Variance and Equal Variance Assumption

When population variances are assumed equal, the sample variances are combined into a pooled variance estimate. This pooled variance provides a more accurate estimate of the common variance, which is used to calculate the standard error of the difference between means in the t-test.

Significance Level (α) and Hypothesis Testing

The significance level α (here 0.01) defines the threshold for rejecting the null hypothesis. If the calculated t-statistic falls in the critical region beyond the t-distribution cutoff, the null hypothesis (no difference in means) is rejected, indicating a statistically significant difference.