How to quit smoking It’s hard for smokers to quit. Perhaps...

How to quit smoking It’s hard for smokers to quit. Perhaps prescribing a drug to fight depression will work as well as the usual nicotine patch. Perhaps combining the patch and the drug will work better than either treatment alone. Here are data from a randomized, double-blind trial that compared four treatments.21 A “success” means that the subject did not smoke for a year following the beginning of the study. (a) Summarize these data in a two-way table. (b) Make a graph to compare the success rates for the four treatments. Describe what you see. (c) Explain in words what the null hypothesis $H_{0} :$: $p_{1}=p_{2}=p_{3}=p_{4}$ says says about subjects’ smoking habits. (d) Find the expected counts if $H_{0}$ is true, and display them in a two-way table similar to the table of observed counts.

How to quit smoking It’s hard for smokers to quit. Perhaps prescribing a drug to fight depression will work as well as the usual nicotine patch. Perhaps combining the patch and the drug will work better than either treatment alone. Here are data from a randomized, double-blind trial that compared four treatments.21 A “success” means that the subject did not smoke for a year following the beginning of the study.
(a) Summarize these data in a two-way table.
(b) Make a graph to compare the success rates for the four treatments. Describe what you see.
(c) Explain in words what the null hypothesis $H_{0} :$: $p_{1}=p_{2}=p_{3}=p_{4}$ says says about subjects’ smoking habits.
(d) Find the expected counts if $H_{0}$ is true, and display them in a two-way table similar to the table of observed counts.

Key Concepts

Null Hypothesis in Categorical Data

In the context of this problem, the null hypothesis asserts that the success rates (or proportions) are equal across all treatment groups. This hypothesis is a baseline assumption used for statistical testing to determine if any treatment has a significantly different effect compared to the others.

Contingency Tables

Contingency tables are a method for summarizing categorical data by displaying the frequencies (or counts) of observations that fall into each combination of categories. They help in understanding the relationship between two categorical variables and are essential for performing subsequent statistical tests such as the chi-square test.

Graphical Comparison of Proportions

Graphical tools like bar charts or mosaic plots are used to visually compare the success rates or proportions across different treatment groups. These graphs make it easier to observe differences or trends among groups, helping to guide further statistical analysis.

Expected Counts Calculation

Expected counts are calculated under the assumption that the null hypothesis is true. They represent the frequency of outcomes that would be expected in each cell of the contingency table if there were no real differences between the groups. These counts are used in tests such as the chi-square test to compare against the observed frequencies.

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