Identifying a Test In Exercises 21–24, determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed.


Ha: μ ≤ 8.0
H0: μ > 8.0

Verified step by step guidance

1. Step 1: Understand the null hypothesis (H₀) and the alternative hypothesis (Hₐ). In this problem, H₀: μ > 8.0 and Hₐ: μ ≤ 8.0. The null hypothesis represents the default assumption, while the alternative hypothesis represents the claim being tested.
2. Step 2: Identify the direction of the inequality in the alternative hypothesis (Hₐ). Here, Hₐ: μ ≤ 8.0 indicates that the mean (μ) is less than or equal to 8.0.
3. Step 3: Recall that the direction of the test is determined by the alternative hypothesis. Specifically: (a) If Hₐ uses '<', it is a left-tailed test. (b) If Hₐ uses '>', it is a right-tailed test. (c) If Hₐ uses '≠', it is a two-tailed test.
4. Step 4: Since Hₐ: μ ≤ 8.0 includes the '≤' symbol, this implies a left-tailed test because we are testing for values less than or equal to a specific value (8.0).
5. Step 5: Conclude that the hypothesis test is left-tailed based on the direction of the inequality in the alternative hypothesis.

Key Concepts

Hypothesis Testing

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

Types of Hypothesis Tests

Hypothesis tests can be classified as left-tailed, right-tailed, or two-tailed based on the direction of the alternative hypothesis. A left-tailed test is used when Ha indicates that the parameter is less than a certain value, while a right-tailed test is used when Ha indicates it is greater. A two-tailed test is employed when Ha suggests that the parameter is simply different from a certain value, without specifying a direction.

Interpreting Hypotheses

In the given hypotheses, Ha: μ ≤ 8.0 suggests that the population mean is less than or equal to 8.0, while H0: μ > 8.0 indicates that the mean is greater than 8.0. This setup implies a left-tailed test, as we are interested in determining if the mean is significantly less than 8.0. Understanding the direction of the hypotheses is crucial for selecting the appropriate statistical test and interpreting the results.