In one study of smokers who tried to quit smoking with nicotine patch therapy, 39 were smoking one year after the treatment, and 32 were not smoking one year after the treatment. Use a 0.10 significance level to test the claim that among the smokers who try to quit with the nicotine patch therapy, the majority are smoking a year after the treatment. Do these results suggest that the nicotine patch therapy is ineffective?

Question

Take a shot, what do you think?
Hint: your question involves a binominal distribution; the formula for the standard deviation is sqrt(n*p*q), where n is the sample size, p it the expected probability, and q=1-p.
Also, you will need a cumulative normal distribution table, probably in the back of your stats text.

Answer 1

To test the claim, we will perform a hypothesis test.

Step 1: State the null hypothesis (H₀) and alternative hypothesis (H₁).
H₀: p ≤ 0.5 (The proportion of smokers who continue smoking one year after treatment is less than or equal to 50%)
H₁: p > 0.5 (The proportion of smokers who continue smoking one year after treatment is greater than 50%)

Step 2: Find the test statistic.
We will use the formula for the test statistic in this case:
z = (p̂ - p) / √(p(1 - p) / n)
where p̂ is the sample proportion, p is the hypothesized proportion, and n is the sample size.

In this case, p̂ = 39/71, p = 0.5, and n = 71. Plug these values into the formula:

z = (39/71 - 0.5) / √(0.5(1 - 0.5) / 71)
z ≈ 0.548 / 0.059
z ≈ 9.29

Step 3: Find the p-value.
Using a standard normal table or calculator, we can find the corresponding p-value for our test statistic, z = 9.29.

p-value ≈ 0

Step 4: Make a decision.
Since our p-value is less than our significance level (α = 0.10), we reject the null hypothesis.

Conclusion:
These results suggest that among the smokers who try to quit with the nicotine patch therapy, the majority are smoking a year after the treatment. However, this does not necessarily mean that the nicotine patch therapy is completely ineffective, as there might be other factors affecting the result. Further research is needed to determine the effectiveness of the therapy.