Question

Consider a manufacturing process that produces cylindrical component parts for the automotive industry. According to specifications, it is important that the process produces parts having a mean diameter of 5.0 millimeters. An experiment is conducted in which 100 parts produced by the process are selected randomly and the diameter measured on each part. It is known that the population standard deviation is 0.1. It was found that the sample mean diameter is 5.027 millimeters. The process engineer Mr. Tan would like to find out how likely is it that one could obtain a sample mean diameter of at least 5.027 with sample size n = 100, if the population mean µ = 5.0. Apply the concept of central limit theorem. Mr Tan claimed that “In only 7 in 1000 experiments, one would experience by chance a sample mean that deviates from the population mean by as much as 0.027.” Do you agree? Explain your reasoning.

Answer 1

Hypothesized mean, = 5.0

Sample mean, = 5.027

Standard error of mean, SE = = 0.1 / = 0.01

Test statistic, Z = ( - ) / SE = (5.027 - 5.0) / 0.01 = 2.7

Using the the concept of central limit theorem, the sampling distribution of sample mean is a Normal distribution.

Probability that one could obtain a sample mean diameter of at least 5.027 with sample size n = 100, if the population mean µ = 5.0 = P(z > 2.7)

= 0.0035

In 1000 experiments, number of samples where a sample mean deviates from the population mean by as much as 0.027 is 1000 * 0.0035 = 3.5

Thus, only 3.5 in 1000 experiments, one would experience by chance a sample mean that deviates from the population mean by as much as 0.027. So, we do not agree with Mr Tan.