# Sampling Distribution and Population Distribution

I assume my scenario will have to include enough information to answer the questions below so I will need help with this one as well

Create and write your own scenario about a specific population. Then, answer the following questions about your scenario:

What are the mean, standard deviation, and shape of the population distribution?

What are the mean, standard deviation and shape of the sample?

What are the mean, standard error and shape of the sampling distribution of the sample mean?

Is it possible for the shapes of these three distributions to be different? Describe a brief scenario where this may happen.

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#### Solution Preview

Note: this is a descrption of scenario only. You need to use the software to get excel values of mean and standard deviaiton. I do not have any software now.

Scebario: step 1: suppose we flip a coin 1000 times. Let head=1 and tail=0. So now we record the number of heads and tails. The population includes 1000 values with either 0 or 1. Then we calculate the mean, standard deviation of the population with 1000 data using software. The shape of the population ...

#### Solution Summary

The solution gives deep discussion on population distribution, sample distribution and sampling distribution of the sample mean. Central limit theorem plays an important role in our analysis.

Sampling Distribution, Mean and Standard Deviation

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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.