Finding Critical Values
In Exercises 17–20, refer to the information in the given exercise and use a 0.05 significance level for the following.


a. Find the critical value(s).
b. Should we reject H0 or should we fail to reject H0?


Exercise 14

Verified step by step guidance

1. Step 1: Identify the type of hypothesis test being conducted (e.g., one-tailed or two-tailed). This is determined by the alternative hypothesis (H1). For example, if H1 states that a parameter is greater than or less than a value, it is a one-tailed test. If H1 states that a parameter is not equal to a value, it is a two-tailed test.
2. Step 2: Determine the degrees of freedom (if applicable). For example, in a t-test, the degrees of freedom are typically calculated as df = n - 1, where n is the sample size.
3. Step 3: Use the significance level (α = 0.05) and the type of test (one-tailed or two-tailed) to find the critical value(s) from the appropriate statistical table (e.g., z-table, t-table, or chi-square table). For a two-tailed test, divide α by 2 to find the critical values for each tail.
4. Step 4: Compare the test statistic (calculated from the sample data) to the critical value(s). If the test statistic falls in the critical region (beyond the critical value(s)), reject the null hypothesis (H0). Otherwise, fail to reject H0.
5. Step 5: State the conclusion in the context of the problem. For example, if H0 is rejected, explain what this means in terms of the original claim or research question.

Key Concepts

Critical Values

Critical values are the threshold points that define the boundaries for rejecting the null hypothesis in hypothesis testing. They are determined based on the significance level (alpha), which indicates the probability of making a Type I error. For a significance level of 0.05, critical values can be found using statistical tables or software, depending on the test type (e.g., z-test, t-test). These values help in deciding whether the test statistic falls into the rejection region.

Null Hypothesis (H0)

The null hypothesis (H0) is a statement that there is no effect or no difference, and it serves as the default assumption in hypothesis testing. Researchers aim to gather evidence against H0 to support an alternative hypothesis (H1). The decision to reject or fail to reject H0 is based on the comparison of the test statistic to the critical values. Understanding H0 is crucial for interpreting the results of statistical tests.

Significance Level (α)

The significance level (α) is the probability threshold set by the researcher before conducting a hypothesis test, commonly set at 0.05. It represents the risk of rejecting the null hypothesis when it is actually true (Type I error). The significance level helps determine the critical values and the rejection region for the test statistic. A lower α indicates a stricter criterion for rejecting H0, while a higher α allows for more leniency.