In Exercises 17 and 18, (a) identify the claim and state Ho and Ha, 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. Step 1: Identify the claim and state the null hypothesis (H\_0) and the alternative hypothesis (H\_a). The claim is that there is no difference between the mean household incomes of the two neighborhoods. Therefore, the null hypothesis is H\_0: \mu_1 = \mu_2, and the alternative hypothesis is H\_a: \mu_1 \neq \mu_2, where \mu_1 and \mu_2 are the population means of the first and second neighborhoods, respectively.
2. Step 2: Note the given sample statistics: For the first neighborhood, sample size n_1 = 12, sample mean \bar{x}_1 = 52750, and sample standard deviation s_1 = 2900. For the second neighborhood, sample size n_2 = 10, sample mean \bar{x}_2 = 51200, and sample standard deviation s_2 = 2225. The significance level is \alpha = 0.01.
3. Step 3: Since the population variances are assumed equal, calculate the pooled sample variance (s_p^2) using the formula: \(s\_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}\)
4. Step 4: Calculate the test statistic t using the formula for two independent samples with equal variances: \(t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}\)
5. Step 5: Determine the degrees of freedom, which is \(df = n_1 + n_2 - 2\). Then, find the critical t-value(s) from the t-distribution table for a two-tailed test at significance level \alpha = 0.01. Compare the calculated t statistic with the critical t-value(s) to decide whether to reject or fail to reject the null hypothesis.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a claim about a population parameter. It involves stating a null hypothesis (Ho) representing no effect or difference, and an alternative hypothesis (Ha) representing the claim to be tested. The test evaluates sample data to determine if the observed effect is statistically significant at a chosen significance level (α).

Two-Sample t-Test for Means with Equal Variances

This test compares the means of two independent samples to determine if there is a significant difference between their population means, assuming equal population variances. It uses a pooled estimate of variance and calculates a t-statistic based on sample means, standard deviations, and sizes. The test is appropriate when samples are random, independent, and populations are normally distributed.

Significance Level and Decision Rule

The significance level (α) defines the threshold for rejecting the null hypothesis, commonly set at 0.01 or 0.05. It represents the probability of making a Type I error—rejecting a true null hypothesis. After calculating the test statistic, it is compared to critical values from the t-distribution; if the statistic falls in the rejection region, the null hypothesis is rejected, supporting the alternative claim.