Question

A manufacturing process is designed to produce bolts with a diameter of 0.5 inches. Once each day, a random sample of 36 bolts is selected and the bolt diameters are recorded. If the resulting sample mean is less than 0.49 inches or greater than 0.51 inches, the process is shut down for adjustment. The standard deviation of bolt diameters is 0.02 inches. What is the probability that the manufacturing line will be shut down unnecessarily? (Hint: Find the probability of observing an \(\bar{x}\) in the shutdown range when the actual process mean is 0.5 inches.)

Short answer

So, the probability that the manufacturing line will be shut down unnecessarily is approximately 0.0026 or 0.26%.

Step-by-step solution

Identify given parameters

The problem gives us several parameters for a manufacturing process. The desired diameter of the bolts is 0.5 inches, the sample size each day is 36 bolts, tolerances for shutting down the process are 0.49 and 0.51 inches, and the standard deviation is 0.02 inches.

Calculate Standard Error

The standard error of the mean (SE) is a measure of how spread out the sample means are around the population mean. It is obtained by dividing the standard deviation by the square root of the sample size. Mathematically, it is expressed as: \(SE = \dfrac{\sigma}{\sqrt{n}}\) where \(\sigma\) is the standard deviation and \(n\) is the sample size. Substituting the given values, we get \(SE = \dfrac{0.02}{\sqrt{36}} = 0.003333\).

Convert to Z-scores

Z-score is a measure of how many standard deviations an element is from the mean. To use the standard normal distribution, we transform the given X values (0.49 inches for the lower limit and 0.51 inches for the upper limit) to z-scores. The Z-score is calculated using formula: \(Z = \dfrac{X - \mu}{SE}\) where X is the value, \(\mu\) is the mean and SE is the standard error. Substituting the means and SE with given values: \(Z_1 = \dfrac{0.49 - 0.5}{0.003333} = -3\) and \(Z_2 = \dfrac{0.51 - 0.5}{0.003333} = 3\)

Calculate Probability

The shutdown will occur if the mean diameter is less than 0.49 (Z1 < -3) or more than 0.51 (Z2 > 3). It means we need to find the probability for the Z-score lying beyond the range of -3 and 3. The probability for Z-score between -3 and 3 (P(-3 < Z < 3)) can be obtained from the standard normal distribution table which is approximately 0.9974. The probability for Z-score lying beyond the range of -3 and 3 = 1 - P(-3 < Z < 3) = 1 - 0.9974 = 0.0026.