# population and standard deviation

6.16

A population has distribution of unknown shape. The mean of the population is 3,500, and the standard deviation is 600.

a. If a sample of 100 values is selected randomly from this population, what is the probability that the sample mean will exceed 3,600?

b. If a sample of 200 is selected from the population what is the probability that the sample mean will exceed 3,600?

c. Compare your answer to parts a and b and explain why the two probabilities are different.

6.11

A population is normally distributed, with a mean of 1,000 and a standard deviation equal to 200.

a. Determine the probability that a random sample of size 5 selected from this population will have a sample mean less than 970?

b. Referring to part a, suppose a second sample of size 10 is selected. What is the probability that this sample will have a mean that is less than 970?

c. Why are the answers to parts a and b different? Discuss

6.17

A random sample of 100 items is selected from a population of size 350. What is the probability that the sample mean will exceed 200 if the population mean is 195 and the population standard deviation equals 20? Hint, use the finite correction factor, because the sample size is more than 5% of the population size!

https://brainmass.com/statistics/sampling/population-and-standard-deviation-142201

#### Solution Summary

Population and standard deviation are discussed.

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.