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.


Annual Salary An employment information service claims the mean annual salary for senior level statisticians is more than \$124,000. The annual salaries (in dollars) for a random sample of 12 senior level statisticians are shown in the table at the left. At α=0.01, is there enough evidence to support the claim that the mean salary is more than \$124,000?

Verified step by step guidance

1. Step 1: Identify the claim and state the null hypothesis (H0) and alternative hypothesis (Ha). The claim is that the mean annual salary for senior level statisticians is more than \$124,000. Thus, H0: μ = \$124,000 (the mean salary is \$124,000) and Ha: μ > \$124,000 (the mean salary is greater than \$124,000).
2. Step 2: Determine the significance level (α) and find the critical value(s). The significance level is given as α = 0.01. Since this is a one-tailed test (Ha: μ > \$124,000), use a t-distribution table to find the critical t-value corresponding to α = 0.01 and degrees of freedom (df = n - 1, where n is the sample size).
3. Step 3: Calculate the standardized test statistic t. First, compute the sample mean (x̄) and sample standard deviation (s) using the provided salary data. Then, use the formula for the t-test statistic: $t=\frac{\mathrm{x̄}-\mu }{\frac{s}{\sqrt{n}}}$, where μ = \$124,000, n = 12, and s is the sample standard deviation.
4. Step 4: Compare the calculated t-value to the critical t-value. If the calculated t-value falls in the rejection region (greater than the 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 enough evidence to support the claim that the mean annual salary for senior level statisticians is more than \$124,000. 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 based on the significance level (α), which represents the probability of making a Type I error (rejecting H0 when it is true). For a one-tailed test, like in this scenario, the rejection region is located in the tail of the distribution, indicating where the test statistic would be considered extreme enough to reject H0.

Standardized Test Statistic

The standardized test statistic, often denoted as t or z, measures how far the sample statistic is from the null hypothesis value in terms of standard deviations. It is calculated using the sample mean, the hypothesized population mean, and the standard error of the mean. This statistic is crucial for determining whether the observed data falls within the rejection region, thus influencing the decision to reject or fail to reject the null hypothesis.