SOLUTION: In 1992, the FAA conducted 86,991 pre-employment drug tests on job applicants who were to be engaged in safety and security-related jobs, and found that 1143 were positive. A. Con

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Question 333028: In 1992, the FAA conducted 86,991 pre-employment drug tests on job applicants who were to be engaged in safety and security-related jobs, and found that 1143 were positive.
A. Construct a 95% confidence interval for the population proportion of positive drug tests.
B. Why is the normality assumption not a problem, despite the very small value of p?
Answer by jrfrunner(365) About Me  (Show Source):
You can put this solution on YOUR website!
1. compute the sample point estimate Pbar = 1143/86991 = 0.0131
2. compute the standard error: sqrt%28pbar%2A%281-pbar%29%2Fn%29=sqrt%280.0131%2A%281-0.0131%29%2F86991%29=0.000386
3. Identify the critical value for 95% confidence: Z=1.96
4. Compute the 95% confidence interval for pi=population_proportion
pbar -\+ Z*Standard Error = 0.0131-1.96*0.000386, 0.0131+1.96*0.000386
(0.0123, 0.0139)
==
Note: This is a binomial distribution problem
X = number testing positive~Binomial (n,p)=binomial(86991,0.0131)
But since the sample size is large and n*pbar=1143 exceeds 10
this meets the criteria for using the normal approximation to the Binomial.
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This is needed since the Binomial is skewed in the extremes and these requirements limit the Binomial from the extremes, allowing the Normal approximation.