Question

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.

Accepted Answer

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$$ 
eq \$$mu_2. $$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. Calculate the standard error of the difference between the two sample means using the pooled standard deviation:

$$SE = s_p 	imes \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}$$ 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. 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.

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Key Concepts

Two-Sample t-Test for Means

Pooled Variance and Equal Variance Assumption

Significance Level (α) and Hypothesis Testing