In one study of smokers who tried to quit smoking with nicotine patch therapy, 36 were smoking after one year after the treatment, and 34 were not smoking one year after the treatment. Use a 0.10 significance level to test the claim that among smokers who try to quit with nicotine patch therapy, the majority are smoking a year after the treatment?
To test the claim that the majority of smokers who try to quit with nicotine patch therapy are smoking a year after the treatment, you can use a hypothesis test. The null hypothesis, denoted as H0, assumes that the majority are not smoking after one year, while the alternative hypothesis, denoted as Ha, assumes that the majority are smoking after one year.
Let's define the proportions:
p = proportion of smokers who are smoking after one year
q = proportion of smokers who are not smoking after one year
Based on the given information, we have:
Number of smokers smoking after one year (successes) = 36
Number of smokers not smoking after one year (failures) = 34
Total sample size = 36 + 34 = 70
We can calculate the sample proportion as:
p̂ = successes / total sample size = 36 / 70 = 0.514
To test the claim, we will conduct a one-sample proportion z-test.
The z-test statistic can be calculated using the formula:
z = (p̂ - p) / sqrt((p * q) / n)
p = proportion assuming the null hypothesis (0.5 for the majority)
q = 1 - p (0.5 in this case)
n = total sample size (70)
Plugging in the values:
z = (0.514 - 0.5) / sqrt((0.5 * 0.5) / 70)
Calculating this, the z-value is approximately 0.434.
Next, we need to determine the critical z-value at the given significance level of 0.10. Since we are performing a one-tailed test (testing for the majority smoking), we need to find the critical z-value that corresponds to an area of 0.90 in the upper tail.
Using a z-table or a statistical calculator, the critical z-value is approximately 1.28 for a one-tailed test with a significance level of 0.10.
Now we can compare the calculated z-value with the critical z-value. If the calculated z-value is greater than the critical z-value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, the calculated z-value (0.434) is less than the critical z-value (1.28). Therefore, we fail to reject the null hypothesis.
Hence, based on the given data, there is not enough evidence to support the claim that the majority of smokers who try to quit with nicotine patch therapy are smoking a year after treatment at a significance level of 0.10.