The State Department of Weights and Measures is responsible for making sure that commercial weighing and measuring devices, such as scales, are accurate so customers and businesses are not cheated. Periodically, employees of the department go to businesses and test their scales. For example, a dairy bottles milk in 1-gallon containers. Suppose that if the filling process is working correctly, the mean volume of all gallon containers is 1.00 gallon with a standard deviation equal to 0.10 gallon. The department's test process requires that they select a random sample of n = 9 containers. If the sample mean is less than 0.97 gallons, the department will fine the dairy. Based on this information, what is the probability that the dairy will get fined even when its filling process is working correctly?
A) 0.90
B) Approximately 0.3159
C) About 0.1841
D) Approximately 0.3821

Question

Quizplus · Accepted Answer

To find the probability that the dairy will get fined even when its filling process is working correctly, we use the standard normal distribution. The mean volume is 1.00 gallon, and the standard deviation is 0.10 gallon. For a sample of n = 9, the standard error of the mean is 0.10 / sqrt(9) = 0.0333 gallons. We calculate the z-score for a sample mean of 0.97 gallons as (0.97 - 1.00) / 0.0333 = -0.9. Looking up this z-score in a standard normal distribution table or using a calculator gives a probability of about 0.1841. This means there's approximately an 18.41% chance the dairy will be fined even if the filling process is working correctly.

The State Department of Weights and Measures Is Responsible for Making