Question

Suppose a group of 700 smokers (who all wanted to give up smoking) were randomly assigned to receive a newly developed drug or a placebo for eight weeks. Of the 380 patients who received the new drug, 52 were not smoking one year later. Of the 320 patients who received the placebo, 85 were not smoking one year later. Does this evidence indicate that the new drug is significantly better at helping people stop smoking? Use a significance level of 5%. Find the z stat, p-value , what is your decision?

Answer 1

The p-value (< 0.0001) is less than the significance level (0.05), we reject the null hypothesis.

Hypothesis Testing: New Drug vs Placebo for Smoking Cessation

Step 1: Set up the null and alternative hypotheses.

Null hypothesis (H0): There is no significant difference in the success rates of quitting smoking between the drug and placebo groups.

Alternative hypothesis (Ha): The success rate of quitting smoking is significantly higher in the drug group compared to the placebo group.

Step 2: Calculate the sample proportions and standard errors.

Success rate for drug group: 52/380 = 0.137

Success rate for placebo group: 85/320 = 0.266

Combined standard error: sqrt((0.137*(1-0.137))/380 + (0.266*(1-0.266))/320) = 0.026

Step 3: Calculate the test statistic (z-score).

z = (0.137 - 0.266) / 0.026 = -4.92

Step 4: Find the p-value.

Using a z-table or statistical software, the p-value for a z-score of -4.92 is less than 0.0001 (highly significant).

Step 5: Make a decision based on the p-value and significance level.

Since the p-value (< 0.0001) is less than the significance level (0.05), we reject the null hypothesis.

This means there is strong evidence to conclude that the new drug is significantly more effective in helping people quit smoking compared to the placebo.