Question

 1. The sampling distribution of the sample mean

 Structural steel is often treated with a protective coating (called an intumescent coating) that insulates the steel against damage from fire. Coatings are rated according to the number of hours of high-temperature exposure the treated steel can withstand, but if the coating is not applied with a mean thickness within the range specified by its manufacturer, the protection it provides will last for a shorter time than its rating.

 An office tower under construction in Boston has been treated with an intumescent coating. Let x = the thickness of the coating (after it has dried) at a point on the building's structural steel. As shown below, the random variable x is uniformly distributed between a = 20 and b = 40 mils (1 mil = 0.001 inch). However, this information, as well as the mean and standard deviation of x, is unknown to the Boston building inspector.

 The Boston building inspector takes thickness readings of the dried intumescent coating at 30 randomly selected points on the building's structural steel parts. The 30 thickness readings constitute a sample of size n = 30, and the information in the sample is used to obtain an estimate of the mean thickness of the building's protective coating. The estimate is used to conduct a hypothesis test that the steel can withstand at least two hours of high temperatures from a fire, as required by the city building code.

 The population of interest is all the points on the building's structural steel. Assume that the population is infinitely large, so an infinite number of samples can be drawn from the population. The sample data for 15 simple random samples (of size n = 30) that could be pulled from the population are in the data set Samples in the Dataview tool. Each sample value is a random draw from a uniform distribution with a = 20 and b = 40.

 Suppose the building inspector happens to select sample 10. The sample data for sample 10 are stored in the variable Sample 10. Use the

 sample mean x as an estimator of the population mean u. The point estimate of μ is _______ 

 Now suppose that instead of sample 10, the inspector had drawn sample 12. Then the point estimate of u would be _______ 

 Selecting a random sample is an example of a random experiment, and the sample statistic x is a numerical description of the result of the experiment. Therefore, x is a random variable. The probability distribution of x is called the sampling distribution of x.

 In practice, you select one random sample and use the information from that sample to estimate the population parameter of interest. However, statisticians sometimes perform a procedure called repeated sampling, in which the experiment is run over and over again and the value of the sample statistic from each running of the experiment is recorded. The distribution of the sample statistics from the repeated sampling is an approximation of the sample statistic's sampling distribution.

 Repeated sampling is used to develop an approximate sampling distribution for x when n = 30. Suppose 1,000 simple random samples (each of size n = 30) are drawn from a uniform population with a = 20 and b = 40. The sample means for the 1,000 samples are located in the Means data set in the variable Sample Mean. (The 15 samples in the Samples data set correspond to the first 15 sample means in the Means data set.)

 

Answer 1