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# Sampling Distribution of the mean

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The amount of time a bank teller spends with each customer has a population mean = 3.1 minutes and population standard deviation = 0.4 minute.
a) What is the probability that for a randomly selected customer the service time would exceed 3 minutes?
b) If many samples of 64 were selected, what are mean and standard error of the mean (standard deviation of sample means) expected to be? What is expected to be the shape of the distribution of sample means? Give reasons for all your answers.
c) If a random sample of 64 customers is selected, what is the probability that the sample mean would exceed 3 minutes?

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#### Solution Preview

The solution is in the attached word file:
The amount of time a bank teller spends with each customer has a population mean M x = 3.1 minutes and population standard deviation sx = 0.4 minute.

a) What is the probability that for a randomly selected customer the service time would exceed 3 minutes?

Mean=M = 3.1 minutes
Standard deviation =s= 0.4 minutes
x= 3 minutes
z=(x-M )/s= -0.25 =(3-3.1)/0.4
Cumulative Probability corresponding to z= -0.25 is= 0.4013
Or Probability corresponding to x< 3.00 is Prob(Z)= 0.4013 0r= 40.13%
Therefore probability corresponding to x> 3.00 is ...

#### Solution Summary

Probability for sample mean is calculated using Central Limit Theorem.

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## 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 &#963; 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 &#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 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 &#956; = 106 by no more than 4.

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