For Exercises 5 through 20, perform each of the following...

For Exercises 5 through $20,$ perform each of the following steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Youth Smoking Researchers suspect that $18 \%$ of all high school students smoke at least one pack of cigarettes a day. At Wilson High School, a randomly selected sample of 300 students found that 50 students smoked at least one pack of cigarettes a day. At $\alpha=0.05$ test the claim that less than $18 \%$ of all high school students smoke at least one pack of cigarettes a day. Use the $P$ -value method.

For Exercises 5 through $20,$ perform each of the following steps.
a. State the hypotheses and identify the claim.
b. Find the critical value(s).
c. Compute the test value.
d. Make the decision.
e. Summarize the results.
Use the traditional method of hypothesis testing unless otherwise specified.
Youth Smoking Researchers suspect that $18 \%$ of all high school students smoke at least one pack of cigarettes a day. At Wilson High School, a randomly selected sample of 300 students found that 50 students smoked at least one pack of cigarettes a day. At $\alpha=0.05$ test the claim that less than $18 \%$ of all high school students smoke at least one pack of cigarettes a day. Use the $P$ -value method.

Key Concepts

Hypothesis Testing

Hypothesis testing is a systematic method used in statistics to decide whether there is enough evidence in a sample accepted to support a particular belief about a population. It involves making assumptions (hypotheses) about a population parameter and then using sample data to either reject or fail to reject those hypotheses.

Null Hypothesis and Alternative Hypothesis

The null hypothesis (H0) is a statement of no effect or no difference, serving as the default assumption to be tested, while the alternative hypothesis (H1) represents the assertion that directly opposes the null hypothesis, reflecting the claim or the new theory. The identification of these hypotheses is crucial as it establishes the framework around which the test is conducted.

Claim Identification

Identifying the claim in hypothesis testing involves determining which hypothesis accurately represents the assertion that is being investigated. This is important because the claim guides the formulation of the hypotheses, particularly indicating which direction the test should be (e.g., less than, greater than, or not equal to a particular value).

Significance Level

The significance level, denoted as alpha (?), is the probability threshold at which the null hypothesis will be rejected. It reflects the researcher’s acceptable risk of committing a Type I error, which is rejecting the null hypothesis when it is actually true. Common significance levels include 0.05, 0.01, and 0.10.

Test Statistic

A test statistic is a value computed from sample data that is used to decide whether to reject the null hypothesis. It quantifies the difference between the observed sample statistic and what is expected under the null hypothesis, factoring in the standard error of the statistic.

Critical Value

The critical value is a point (or points) on the test statistic distribution that defines the threshold or boundaries of the rejection region. If the test statistic falls in the rejection region (beyond the critical value), the null hypothesis is rejected.

P-value Method

The p-value is the probability of obtaining a test statistic at least as extreme as the one computed from the sample data, assuming that the null hypothesis is true. In the p-value method, the null hypothesis is rejected if the p-value is less than or equal to the chosen significance level, providing a measure of the evidence against the null hypothesis.

Decision and Conclusion

The decision stage in hypothesis testing involves comparing the p-value or test statistic with the critical value to determine whether to reject the null hypothesis. The conclusion summarizes the statistical findings in the context of the original claim, clarifying what the results suggest about the population parameter under investigation.

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