Efficacy of e-Cigs Do electronic cigarettes assist in helping individuals quit smoking? Researchers found 300 current smokers to volunteer for a study in which each was randomly assigned to one of three treatment groups. Group 1 received an electronic cigarette (e-cig) in which each cartridge contained 7.2 mg of nicotine, Group 2 received an e-cig that contained 5.4 mg of nicotine, and Group 3 received an e-cig that contained no nicotine. The subjects did not know which group they were assigned. During the course of the 52-week intervention, subjects dropped out of the study. At the end of the study 65 subjects remained in Group 1, 63 in Group 2, and 55 in Group 3. After 52 weeks, it was determined via questionnaire whether the subject quit smoking entirely. Results of the study are presented in the following table.

g. Write a conclusion for this hypothesis test.

Verified step by step guidance

1. Step 1: Define the hypotheses. The null hypothesis (H0) states that the proportion of individuals who quit smoking is the same across all three groups. The alternative hypothesis (Ha) states that at least one group has a different quitting proportion.
2. Step 2: Calculate the sample proportions of quitting for each group. For each group, divide the number of individuals who quit by the total number of individuals remaining in that group (e.g., for Group 1, calculate \(\hat{p}_1 = \frac{13}{65}\)).
3. Step 3: Use a chi-square test for homogeneity to compare the quitting rates across the three groups. Construct a contingency table with the observed counts (Quit and Did Not Quit for each group) and calculate the expected counts under the null hypothesis.
4. Step 4: Compute the chi-square test statistic using the formula \(\chi^2 = \sum \frac{(O - E)^2}{E}\), where \(O\) is the observed count and \(E\) is the expected count for each cell in the contingency table.
5. Step 5: Determine the degrees of freedom for the test, which is \((\text{number of rows} - 1) \times (\text{number of columns} - 1)\), and compare the test statistic to the critical value from the chi-square distribution or calculate the p-value. Based on this, decide whether to reject or fail to reject the null hypothesis and write a conclusion about whether the nicotine content in e-cigarettes affects quitting rates.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis. It involves formulating a null and alternative hypothesis, calculating a test statistic, and comparing it to a critical value or p-value to draw conclusions about the population.

Chi-Square Test for Independence

The chi-square test for independence assesses whether two categorical variables are related. In this context, it tests if quitting smoking is independent of the treatment group. It compares observed frequencies with expected frequencies under the assumption of independence.

Random Assignment and Blinding

Random assignment ensures each participant has an equal chance of being in any group, reducing bias and confounding variables. Blinding means subjects do not know their group, preventing placebo effects and bias in reporting outcomes, which strengthens the validity of the study results.