Test the given claim. Identify the null hypothesis,...

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Among 198 smokers who underwent a "sustained care" program, 51 were no longer smoking after six months. Among 199 smokers who underwent a "standard care" program, 30 were no longer smoking after six months (based on data from "Sustained Care Intervention and Post discharge Smoking Cessation Among Hospitalized Adults," by Rigotti et al., Journal of the American Medical Association, Vol. 312, No. 7). We want to use a 0.01 significance level to test the claim that the rate of success for smoking cessation is greater with the sustained care program. a. Test the claim using a hypothesis test. b. Test the claim by constructing an appropriate confidence interval. c. Does the difference between the two programs have practical significance?

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim.
Among 198 smokers who underwent a "sustained care" program, 51 were no longer smoking after six months. Among 199 smokers who underwent
a "standard care" program, 30 were no longer smoking after six months (based on data from "Sustained Care Intervention and Post discharge Smoking Cessation Among Hospitalized Adults," by Rigotti et al., Journal of the American Medical Association, Vol. 312, No. 7). We want to use a 0.01 significance level to test the claim that the rate of success for smoking cessation is greater with the sustained care program.
a. Test the claim using a hypothesis test.
b. Test the claim by constructing an appropriate confidence interval.
c. Does the difference between the two programs have practical significance?

Key Concepts

Confidence Interval

A confidence interval is an estimated range of values that is likely to include the true difference between the population parameters with a specified level of confidence. If the interval does not contain the null value (often zero), it suggests statistical significance.

Null Hypothesis

This is the foundational assumption in hypothesis testing that there is no effect or no difference between groups. It establishes a baseline against which any observed effect is measured and is assumed true until evidence suggests otherwise.

Alternative Hypothesis

This hypothesis represents the claim or effect that the researcher intends to support — typically, it states that there is an effect or a difference in a particular direction. It is accepted if the evidence is strong enough to reject the null hypothesis.

Two-Proportion Z-Test

A two-proportion z-test is used to compare the proportions from two independent samples to determine whether there is a statistically significant difference between them. It is appropriate when the sample sizes are sufficiently large for the normal approximation to hold.

Test Statistic

The test statistic, in the context of proportions, quantifies how far the observed difference between sample proportions is from the hypothesized difference under the null hypothesis, scaled by its standard error. It serves as the basis for determining the significance of the observed result.

Significance Level

The significance level is a threshold set by the researcher (commonly denoted as alpha) to determine when to reject the null hypothesis. It represents the probability of making a Type I error, which is rejecting a true null hypothesis.

P-Value

The P-value is a measure that indicates the probability of observing the test statistic, or one more extreme, assuming that the null hypothesis is true. A small P-value suggests that the observed data is unlikely under the null hypothesis, leading to its potential rejection.

Practical Significance

Practical significance assesses the real-world importance or magnitude of the observed effect, beyond just statistical significance. It considers whether the difference is large enough to have meaningful implications or to drive decision-making in a practical context.

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