Sampling Distribution, Mean and Standard Deviation

See attachment for better symbol representation.

1) A manufacturer of paper used for packaging requires a minimum strength of 20 pounds per square inch. To check on the quality of the paper, a random sampleof 10 pieces of paper is selected each hour from the previous hour’s production and a strength measurement is recorded for each. The standard deviationσ of the strength measurements, computed by pooling the sum of squares of deviations of many samples, is know to equal 2 pounds per square inch, and the strength measurements are normally distributed.

a) What is the approximate samplingdistribution of the sample meanof n = 10 test pieces of paper?
b) If the mean of the population of strength measurements is 21 pounds per square inch, what is the approximate probabilitythat, for a random sample of n = 10 test pieces of paper, ¯x < 20?
c) What value would you select for the mean paper strength &#956; in order that P (¯x < 20) be equal to .001?

2) Suppose a random sample of n = 25 observations is selected from a population that is normally distributed, with mean equal to 106 and standard deviation equal to 12?
a) Give the mean and standard deviation of the sampling distributionof the sample mean ¯x.
b) Find the probability that ¯x exceeds 110
c) Find the probability that the sample mean deviates from the population mean &#956; = 106 by no more than 4.

This solution shows step-by-step calculations to determine sampling distribution, probabilities of test paper mean, mean paper strength, mean, standard deviation and probabilities of normally distributed population. All workings are shown with brief explanations.