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# Probability Distribution & Standard Deviation

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The New York City Housing and Vacancy Survey showed a total of 59,324 rent-controlled housing units and 236,263 rent-stabilized units built in 194 or later. For these rental units, the probability distributions for the number of persons living in the unit are given.

Number of Persons Rent-Controlled Rent-Stabilized
1 0.61 0.41
2 0.27 0.30
3 0.07 0.14
4 0.04 0.11
5 0.01 0.03
6 0.00 0.01

a. What is the expected value of the number of people living in rent-controlled units?
Rent Controlled: E(x) =

b. What is the expected value of the number of people living in rent-stabilized units?
Rent Stabilized: E(x) =

c. What is the standard deviation of the number of people living in rent-controlled units?
Var(x) =
Standard Deviation =

d. What is the standard deviation of the number of people living in rent-stabilized units?
Var(x) =
Standard Deviation =

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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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