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 sample of 10 pieces of paper is selected each hour from the previous hour's production and a strength measurement is recorded for each. The standard deviation σ 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 sampling distribution of the sample mean of 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 probability that, for a random sample of n = 10 test pieces of paper, ¯x < 20?
c) What value would you select for the mean paper strength μ 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 distribution of 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 μ = 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.

...mean, variance, and standard deviation of the sampling distribution of the ... The standard deviations of sampling distributions of the sample mean is examined. ...

... For example, in Chart 74 the probability distributions of length of ... There is another normal prob- ability distribution for the length of service in the ...

... 0.2913 =291/999. 5 questions on point estimate of mean, proportion and standard deviation, sampling distribution, standard error etc. have been answered. ...

...distributions, n > 15 For normal population distributions, the sampling... 15 we can be reasonably certain that the sampling distribution of the mean will be ...

...deviation 16, then the mean and the standard deviation for the sampling distribution of x-bar for n=16 are: 2) Consider two population distributions labeled A ...

... Find the mean and the standard deviation of the sampling distribution of x, and find an interval containing 95.44% of all possible sample mean returns. ...

... Limit Theorem: Even if the population is not normal, sample means from the ... For most distributions, n > 30 will give a sampling distribution that is nearly ...

... value of the population mean is the mean of sample means c. The Standard error of the mean is the standard deviation of the distribution of sample means. ...

... questions: Mean, median, standard deviation of a frequency distribution, skewness of the data. 1. Owens Orchards sells apples in a large bag by weight. A sample...

... Find the mean of the sampling distribution. 4. Assume that z scores are normally distributed with a mean of 0 and a standard deviation of 1. If P(0 < z < a ...