Exercise 20 Refer to the information in the given exercise...

Exercise 20
Refer to the information in the given exercise and do the following.
a. Find the critical value(s).
b. Using a significance level of $\alpha=0.05$, should we reject $H_{0}$ or should we fail to reject $H_{0}$ ?

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, which represents a default or status quo assumption, and an alternative hypothesis, which represents a competing claim. Through this method, evidence is evaluated to decide whether to reject the null hypothesis in favor of the alternative hypothesis.

Null Hypothesis and Alternative Hypothesis

The null hypothesis (H?) is a statement asserting that there is no effect or no difference in the population, and it serves as the basis for testing. The alternative hypothesis (H? or H?) is a statement that contradicts the null hypothesis, suggesting the presence of an effect or difference. The decision structures in hypothesis testing revolve around these two opposing statements.

Significance Level

The significance level, denoted by ?, is the probability threshold used to decide whether to reject the null hypothesis. It represents the maximum probability of committing a Type I error, which is the error of rejecting a true null hypothesis. Common significance levels include 0.05, 0.01, and 0.10.

Critical Value

A critical value is a point (or points) on the scale of the test statistic beyond which the null hypothesis is rejected. These values define the cutoff regions in the probability distribution of the test statistic under the null hypothesis. Determining the critical value depends on the chosen significance level, the nature of the test (one-tailed or two-tailed), and the sampling distribution of the statistic.

Decision Rule

The decision rule in hypothesis testing is a guideline used to determine whether to reject or fail to reject the null hypothesis based on the computed test statistic and critical value(s). If the test statistic falls in the critical region beyond the critical value(s), the null hypothesis is rejected. Otherwise, there is insufficient evidence to reject it.

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