Pinky Bauer,Chief Financial Officer of Harrison Haulers,Inc.,suspects irregularities in the payroll system. If 10% of the 5,000 payroll vouchers issued since January 1,2000,have irregularities,the probability that Pinky's random sample of 200 vouchers will have a sample proportion of between .06 and .14 is ___________.
A) 0.4706
B) 0.9706
C) 0.0588
D) 0.9412
E) 0.8765

Question

Quizplus · Accepted Answer

The problem describes a binomial probability distribution where the probability of success (having an irregularity in a voucher) is 0.1 and the sample size is 200. We need to find the probability that the sample proportion (p̂) falls between 0.06 and 0.14.

First, we find the mean and standard deviation of the distribution using the formula:

μ = np = 200 x 0.1 = 20 
σ = sqrt(np(1-p)) = sqrt(200 x 0.1 x 0.9) = 4.74

Next, we standardize the sample proportion values using the z-score formula:

z1 = (0.06 - 0.1) / 0.0474 = -0.84 
z2 = (0.14 - 0.1) / 0.0474 = 0.84

Using a standard normal distribution table or calculator, we find the area between these two z-scores to be approximately 0.9412 or 94.12%. Therefore, the answer is D.