Question

In 2002 the Supreme Court ruled that schools could require random drug tests of students participating in competitive after-school activities such as athletics. Does drug testing reduce use of illegal drugs? A study compared two similar high schools in Oregon. Wahtonka High School tested athletes at random and Warrenton High School did not. In a confidential survey, 6 of 124 athletes at Wahtonka and 26 of 105 athletes at Warrenton said they were using drugs. Regard these athletes as SRSs from the populations of athletes at similar schools with and without drug testing. (a) You should not use the large-sample confidence interval. Why not? A. The sample sizes are too small. B. The sample sizes are not identical. C. The sample proportions are too small. D. At least one sample has too few failures. E. At least one sample has too few successes. (b) Give the plus four 99.9% confidence interval for the difference between the proportion of athletes using drugs at schools with and without testing using the plus four method. Interval:

Answer 1

The large-sample confidence interval is not appropriate due to small sample proportions. The plus four 99.9% confidence interval for the difference in drug use proportions between the two schools is calculated using the plus four adjusted counts and the formula for the confidence interval of two proportions.

Step-by-step explanation:

In the question presented, two schools are studied for the impact of random drug testing on the use of illegal drugs by athlete students. Let's address part (a) first. Given the sample sizes and the number of athletes reporting drug use, we should not use the large-sample confidence interval because the sample proportions are too small, which is answer C. Large sample confidence intervals assume that the sample sizes are large enough for the Central Limit Theorem to apply, which isn't the case here because the sample sizes and the number of successes/failures don't meet the necessary conditions.

Now for part (b), we can calculate the plus four 99.9% confidence interval for the difference between the proportions. To do this, we add four to the count of successes and four to the count of trials for both samples, giving us adjusted counts of 10 out of 128 for Wahtonka and 30 out of 109 for Warrenton. The plus four method helps stabilize the variance when dealing with small sample sizes and extreme proportions.

The confidence interval formula for two proportions is:

p1 = 10/128, p2 = 30/109

Point estimate, pe = p1 - p2

Z* for 99.9% confidence level is approximately 3.291

Standard error, SE = sqrt[p1(1-p1)/128 + p2(1-p2)/109]

Margin of error, ME = Z* × SE

Confidence interval, CI = pe ± ME

After these calculations, you'll have the 99.9% confidence interval for the difference in drug use proportions.