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Does being part of a support group affect the ability of people to quit smoking? A county health department enrolled 300 smokers in a randomized experiment. 150 participants were assigned to a group that used a nicotine patch and met weekly with a support group; the other 150 received the patch and did not meet with a support group. At the end of the study, 40 of the participants in the patch plus support group had quit smoking while only 30 smokers had quit in the other group. (a) Create a two-way table presenting the results of this study. (b) Answer each of the following questions under the null hypothesis that being part of a support group does not affect the ability of people to quit smoking, and indicate whether the expected values are higher or lower than the observed values. i. How many subjects in the "patch + support" group would you expect to quit? ii. How many subjects in the "patch only" group would you expect to not quit?

Step by step solution

01

Set up the Two-Way Table

To construct a two-way table, we need to summarize the data provided. We have two groups: 'patch + support group' and 'patch only group'. We categorize them based on whether they quit smoking or did not quit smoking. Here's the table:\[\begin{array}{|c|c|c|}\hline & \text{Quit Smoking} & \text{Did Not Quit} \\hline\text{Patch + Support} & 40 & 110 \\hline\text{Patch Only} & 30 & 120 \\hline\end{array}\]The values 110 and 120 are calculated by subtracting 40 and 30 (those who quit) from the total participants in each group (150).

02

Establish Total Counts

We need the total counts to calculate expected frequencies under the null hypothesis. - Total who quit: \(40 + 30 = 70\)- Total who did not quit: \(110 + 120 = 230\)- Grand total: \(300\) (as all participants are accounted for).This helps set the stage for calculating expected values.

03

Calculate Expected Values under Null Hypothesis

Under the null hypothesis, being part of a support group does not affect the quitting rate, implying independent events. The expected frequency for a cell is calculated as follows:e.g., for 'patch + support' group quitting:\[E(quit | \text{patch + support}) = \frac{(150) \cdot (70)}{300} = 35\]Similarly,\[E(\text{not quit} | \text{patch only}) = \frac{(150) \cdot (230)}{300} = 115\]These expected values let us compare expectations with observations.

04

Compare Observed vs. Expected Values

Now that we have the expected values, we can compare them to the observed results: - Observed 'patch + support' who quit: 40 - Expected 'patch + support' who quit: 35 - Observed 'patch only' who did not quit: 120 - Expected 'patch only' who did not quit: 115 Here, observed values for 'patch + support' who quit are higher, and those who did not quit among 'patch only' are lower than expected.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about a population based on sample data. In this exercise, hypothesis testing helps determine if being part of a support group influences quitting smoking.
The hypothesis testing process generally involves these steps:

• Formulate the null and alternative hypotheses.
• Choose a significance level, often denoted as \(\alpha\).
• Calculate the test statistic.
• Compare the test statistic to a critical value or use a p-value to make a decision.

In this study, the question is whether the support group has an effect on quitting smoking. The null hypothesis asserts that the support group makes no difference in the quitting rate, while the alternative hypothesis suggests it does have an effect.

Randomized Experiment

A randomized experiment is a study design that assigns participants to different groups using randomization. This design minimizes bias and allows for the examination of causal relationships.
In this exercise, 300 smokers were randomly assigned to two groups:

• Patch + Support Group: Participants use a nicotine patch and attend support group sessions.
• Patch Only Group: Participants only use a nicotine patch.

The random assignment ensures any differences in outcomes are likely due to the intervention itself (support group) rather than pre-existing differences between groups. This makes any observed differences in quitting rates between the groups more attributable to the support group involvement.

Expected Values

Expected values in statistics are the calculated averages for random variables. Under the null hypothesis, these expected values help in comparing what we would "expect" to happen against the actual observed results.
In a two-way table like in this study:

• The expected number of people who quit in the "patch + support" group is calculated by multiplying the total for that group by the overall proportion who quit: \[E(quit | \text{patch + support}) = \frac{150 \cdot 70}{300} = 35\]
• The expected number who didn't quit in the "patch only" group is similarly calculated: \[E(\text{not quit} | \text{patch only}) = \frac{150 \cdot 230}{300} = 115\]

These expected values serve as a benchmark. By comparing them to observed counts, we can gauge whether the observed differences are large enough to suggest a real effect beyond chance.

Null Hypothesis

The null hypothesis (\(H_0\) in statistics) is a statement asserting that there is no effect or no difference. In the context of this study, the null hypothesis suggests that being part of a support group does not impact the ability to quit smoking.
Establishing a null hypothesis is crucial as it sets a baseline for statistical testing. It's what researchers seek to test against the observed data.
When experimental results deviate significantly from what the null hypothesis predicts, it can be rejected in favor of the alternative hypothesis. However, if results are not substantially different, we "fail to reject" the null hypothesis, supporting it indirectly by default.
A study's conclusion depends on whether the findings show a statistically significant effect, thus providing evidence against the null hypothesis.