Hypothesis Testing Using Rejection Regions In Exercises 19–26, (a) identify the claim and state H0 and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic t, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the population is normally distributed.


Credit Card Debt A credit reporting agency claims that the mean credit card debt in Colorado is greater than \$5540 per borrower. You want to test this claim. You find that a random sample of 30 borrowers has a mean credit card debt of \$5594 per person and a standard deviation of \$597 per person. At , can you support the claim α=0.05?

Verified step by step guidance

1. Step 1: Identify the claim and state the null and alternative hypotheses. The claim is that the mean credit card debt in Colorado is greater than \$5540. This translates to the alternative hypothesis (Ha): μ > 5540. The null hypothesis (H0) is the opposite of the claim: μ ≤ 5540.
2. Step 2: Determine the critical value(s) and rejection region(s). Since this is a one-tailed test (greater than), use the t-distribution table to find the critical t-value for a significance level of α = 0.05 and degrees of freedom (df) = n - 1 = 30 - 1 = 29. The rejection region will be t > critical t-value.
3. Step 3: Calculate the standardized test statistic t. Use the formula: t = (x̄ - μ) / (s / √n), where x̄ is the sample mean (\$5594), μ is the hypothesized population mean (\$5540), s is the sample standard deviation (\$597), and n is the sample size (30). Substitute the values into the formula to compute t.
4. Step 4: Compare the calculated t-value to the critical t-value. If the calculated t-value falls in the rejection region (t > critical t-value), reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
5. Step 5: Interpret the decision in the context of the original claim. If the null hypothesis is rejected, conclude that there is sufficient evidence to support the claim that the mean credit card debt in Colorado is greater than \$5540. If the null hypothesis is not rejected, conclude that there is not enough evidence to support the claim.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents a statement of no effect or no difference, and the alternative hypothesis (Ha), which represents the claim being tested. The goal is to determine whether there is enough evidence in the sample data to reject H0 in favor of Ha.

Rejection Region

The rejection region is a set of values for the test statistic that leads to the rejection of the null hypothesis. It is determined by the significance level (α), which defines the probability of making a Type I error (rejecting H0 when it is true). In hypothesis testing, if the calculated test statistic falls within this region, we reject H0, indicating that the sample provides sufficient evidence to support Ha.

Standardized Test Statistic

The standardized test statistic, often denoted as t or z, measures how far the sample mean is from the population mean under the null hypothesis, expressed in terms of standard deviations. It is calculated using the sample mean, population mean (from H0), sample standard deviation, and sample size. This statistic is crucial for determining whether the observed data is consistent with the null hypothesis or if it suggests a significant difference.