Question

The NCHS report indicated that in 2014 the prevalence of teen smoking was 14.3%. A researcher gathered data on teen smoking from the same year and found about 22.5% of his sample of teens (n=233) smoked. Based on hypothesis testing to detect a difference between the prevalence of smoking in this researcher's sample compared and the NCHS prevalence, which one of the following statements is correct?

A. We fail to reject the null hypothesis at the alpha level of .05.
B. The obtained test statistic value is 3.27.
C. A one-sample test of proportion cannot be conducted to answer this research question, because the assumption, Min[np0, n(1-p0)] equal to or greater than 5, is violated.
D. One should run a one-sample t test.

Answer 1

Explanation:

We would set up the hypothesis test.

For the null hypothesis,

p = 14.3/100 = 0.143

For the alternative hypothesis,

P ≠ 0.143

Considering the population proportion, probability of success, p = 0.143

q = probability of failure = 1 - p

q = 1 - 0.143 = 0.857

Considering the sample,

n = number of samples = 233

P = 22.5/100 = 0.225

We would determine the test statistic which is the z score

z = (P - p)/√pq/n

z = (0.225 - 0.143)/√(0.143 × 0.857)/233 = 3.58

Recall, population proportion, P = 0.143

The difference between sample proportion and population proportion(P - p) is 0.225 - 0.143 = 0.082

Since the curve is symmetrical and it is a two tailed test, the p for the left tail is 0.143 - 0.082 = 0.061

the p for the right tail is 0.143 + 0.082 = 0.225

These proportions are lower and higher than the null proportion. Thus, they are evidence in favour of the alternative hypothesis. We will look at the area in both tails. Since it is showing in one tail only, we would double the area

From the normal distribution table, the area above the z score in the right tail is 1 - 0.99983 = 0.00017

We would double this area to include the area in the left tail of z = - 3.58. Thus,

p = 0.00017 × 2 = 0.00034

Since alpha, 0.05 > than the p value, 0.00034, then we would reject the null hypothesis. Therefore, none of the following statements is correct