Question

A health magazine conducted a survey on the drinking habits of US young adults (ages 21-35). On the question "Do you drink beer, wine, or hard liquor each week?" 985 of the 1516 adults interviewed said "yes".

A. Test at 0.02 significance level whether the population proportion of US young adults who drink beer, wine, or hard liquor on a weekly basis is significantly different from 65%. Include the hypotheses, the Z-test statistic, the p-value, test decision and conclusion in the context of the problem.

B. Do you believe your results are valid? Explain.

C. What test decision error could you have made and provide an explanation of this error in context of the problem.

D. Include a copy of your R-code and test output.

Answer 1

Let be the population proportion of US young adults taking beer, wine, or hard liquor each week.

A) The hypotheses are -

The Z-statistic for test is,

Thus, for given problem,  

As P-value is greater than 0.02 (level of significance) we fail to reject the null hypothesis. Thus, the hypothesis we concerned - whether the proportion of US young adults who drink beer, wine, or hard liquor on a weekly basis is significantly different from 65% is wrong.

(B) The result seems to be valid because the sample proportion was also very close to 0.65. Further, formal statistical testing using the p-value confirmed the fact.

(C) In this situation, we could have made type II error - Accepting the null hypothesis when it is false. In the problem context, this is the error of concluding the proportion is 0.65 when in reality it is significantly different from 0.65.

(D)

p_hat <- 985/1516 Z_cal <- (p_hat - 0.65)/ sqrt(0.65*(1-0.65)/1516) p_value <- 2*pnorm(Z_cal) cat("\nsample proportion: ", p_hat, "\nZ statistic : ", Z_cal, "\nP-value :", p_value); 

Output:

sample proportion: 0.6497361
Z statistic : -0.02153871
P-value : 0.9828159