Take this test as you would take a test in class.For each exercise, perform the steps below.


a. Identify the claim and state and


b.Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed, and whether to use a z-test or a t-test. Explain your reasoning.


c.Find the critical value(s) and identify the rejection region(s).


d. Find the appropriate standardized test statistic.


e. Decide whether to reject or fail to reject the null hypothesis.


f. Interpret the decision in the context of the original claim.


A demographics researcher claims that the mean household income in a recent year is different for native-born households and foreign-born households. A sample of 18 native-born households has a mean household income of \$69,474 and a standard deviation of \$21,249. A sample of 21 foreign-born households has a mean household income of \$64,900 and a standard deviation of \$17,896. At α=0.01, can you support the demographics researcher’s claim? Assume the populations are normally distributed and the population variances are not equal. (Adapted from U.S. Census Bureau)

Verified step by step guidance

1. Step 1: Identify the claim and state the null and alternative hypotheses. The claim is that the mean household income is different for native-born households and foreign-born households. This implies a two-tailed test. The null hypothesis (H₀) is that the means are equal: μ₁ = μ₂. The alternative hypothesis (Hₐ) is that the means are not equal: μ₁ ≠ μ₂.
2. Step 2: Determine the type of test to use. Since the population variances are not equal and the sample sizes are small (n₁ = 18, n₂ = 21), a two-sample t-test is appropriate. This is because the t-test is robust for small sample sizes and unequal variances when the populations are normally distributed.
3. Step 3: Find the critical value(s) and identify the rejection region(s). For a two-tailed test at a significance level of α = 0.01, divide α by 2 to account for both tails (α/2 = 0.005). Use the degrees of freedom formula for unequal variances: df = [(s₁²/n₁ + s₂²/n₂)²] / [(s₁²/n₁)²/(n₁-1) + (s₂²/n₂)²/(n₂-1)]. Once the degrees of freedom are calculated, use a t-distribution table or software to find the critical t-value for α/2 = 0.005. The rejection regions are t < -t_critical and t > t_critical.
4. Step 4: Calculate the standardized test statistic. Use the formula for the t-statistic for two independent samples with unequal variances: t = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂), where x̄₁ and x̄₂ are the sample means, s₁ and s₂ are the sample standard deviations, and n₁ and n₂ are the sample sizes. Plug in the given values to compute the t-statistic.
5. Step 5: Make a decision and interpret the result. Compare the calculated t-statistic to the critical t-value(s). If the t-statistic falls in the rejection region, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Finally, interpret the decision in the context of the claim: if the null hypothesis is rejected, there is sufficient evidence to support the claim that the mean household incomes are different. If the null hypothesis is not rejected, there is insufficient evidence to support the claim.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample data to determine whether there is enough evidence to reject H0 in favor of H1. This process helps researchers assess claims and draw conclusions about populations.

T-test vs. Z-test

The choice between a t-test and a z-test depends on the sample size and whether the population standard deviation is known. A t-test is used when the sample size is small (typically n < 30) and the population standard deviation is unknown, while a z-test is appropriate for larger samples or when the population standard deviation is known. In this case, since the sample sizes are small and the population variances are not equal, a t-test is suitable.

Critical Values and Rejection Regions

Critical values are the threshold points that define the boundaries of the rejection region in hypothesis testing. They are determined based on the significance level (α) and the type of test (one-tailed or two-tailed). The rejection region is where, if the test statistic falls, we reject the null hypothesis. For this problem, with α = 0.01, identifying the critical values will help determine if the observed test statistic supports the researcher's claim.