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 closest to___.
A)0.4706
B)0.9706
C)0.0588
D)0.9412
E)0.8765

Question

Quizplus · Accepted Answer

The proportion of vouchers with irregularities in the entire population is p=0.10. 
The sample size is large enough (n=200) to use the normal approximation to the binomial distribution. 
The mean of the sample proportion is: 
μp = p = 0.10 
The standard deviation of the sample proportion is: 
σp = sqrt((p*(1-p))/n) = sqrt((0.10*0.90)/200) ≈ 0.0224 
Using the z-score formula: 
z = (x - μp)/σp 
For x=0.06: 
z = (0.06 - 0.10)/0.0224 ≈ -1.79 
For x=0.14: 
z = (0.14 - 0.10)/0.0224 ≈ 1.79 
The probability of obtaining a sample proportion between 0.06 and 0.14 is the probability of the standard normal variable Z being between -1.79 and 1.79. This probability can be calculated using a standard normal table or a calculator: 
P(-1.79 < Z < 1.79) ≈ 0.9412 
Therefore, the closest probability to the requested value is D) 0.9412.