A design engineer wants to construct a sample mean chart for controlling the service life of a halogen headlamp his company produces. He knows from numerous previous samples that when this service life is in control it is normally distributed with a mean of 500 hours and a standard deviation of 20 hours. On three recent production batches, he tested service life on random samples of four headlamps, with the results?

Answer:[tex]$ \text {Sample mean} = \bar{x} = \mu = 500 \: hours $[/tex]Step-by-step explanation:What is Normal Distribution?We are given a Normal Distribution, which is a continuous probability distribution and is symmetrical around the mean. The shape of this distribution is like a bell curve and most of the data is clustered around the mean. The area under this bell shaped curve represents the probability.  For the given scenario, it is known from numerous previous samples that when this service life is in control it is normally distributed with a mean of 500 hours and a standard deviation of 20 hours. On three recent production batches, he tested service life on random samples of four headlamps.We are asked to find the mean of the sampling distribution of sample means when the service life is in control.Since we know that the population is normally distributed and a random sample is taken from the population then the mean of the sampling distribution of sample means would be equal to the population mean that is 500 hours.[tex]$ \text {Sample mean} = \bar{x} = \mu = 500 \: hours $[/tex]Whereas the standard deviation of the sampling distribution of sample means would be[tex]\text {standard deviation} = s = \frac{\sigma}{\sqrt{n} } \\\\[/tex]Where n is the sample size and σ is the population standard deviation.[tex]\text {standard deviation} = s = \frac{20}{\sqrt{4} } \\\\ \text {standard deviation} = s = \frac{20}{2 } \\\\ \text {standard deviation} = s = 10 \: hours \\\\[/tex]