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# Binomial Random Variable

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23. The 2002 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 1947 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 .61 .41
2 .27 .30
3 .07 .14
4 .04 .11
5 .01 .03
6 .00 .01

A. What is the expected value of the number of persons living in each type of unit?
B. What is the variance of the number of persons living in each type of unit?
C. Make some comparisons between the number of persons living in rent-controlled units and the number of persons living in rent-stabilized units.

27. Consider a binomial experiment with n = 20 and p = .70

A. Compute f(12).
B. Compute f(16)
C. Compute P(x 16).
D. Compute P(x 15).
E. Compute E(x)
F. Compute Var(x) and

29. According to a BusinessWeek/Harris poll of 1035 adults, 40% of those surveyed agreed strongly with the proposition that business has too much power over American life (Business-Week, September 11m 2000). Assume this percentage is representative of the American population. In a sample of 20 individuals taken from a cross-section of the American population, what is the probability that at least five of these individuals will feel that business has too much power over American life?

31. Nine percent of undergraduate students carry credit card balances greater than \$7000. Suppose 10 undergraduate students are selected randomly to be interviewed about credit card usage.
A. Is the selection of 10 students a binomial experiment? Explain
B. What is the probability that two of the students will have a credit card balance greater than \$7000?
C. What is the probability that none will have a credit card balance greater than \$7000?
D.What is the probability that at least 3 will have a credit card balance greater than \$7000?

33. Fifty percent of Americans believed the country was in a recession, even though technically the economy had not shown two straight quarters of negative growth. For a sample of 20 Americans, make the following calculations.
A. Compute the probability that exactly 12 people believed the country was in a recession?
B. compute the probability that no more than five people believed the country was in a recession?
C. How many people would you expect to say the country was in a recession?
D. Compute the variance and standard deviation of the number of people who believed the country was in a recession.

37. Seventy-two percent of Americans have online access. In a random sample of 30 people, what is the expected number of persons with online access? What are the variance and standard deviation?

38. Write a paragraph about the binomial distribution. In the paragraph create a situation (example) that uses the binomial distribution as a model. Explain each of the requirements for a binomial distribution that is listed in the Key Points and explain how your model fits that requirement. Then state how to compute the mean and standard deviation of this distribution.

See attached file for full problem description.

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https://brainmass.com/statistics/probability/binomial-random-variable-112889

#### Solution Preview

Please see the attached file.

23. The 2002 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 1947 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 .61 .41
2 .27 .30
3 .07 .14
4 .04 .11
5 .01 .03
6 .00 .01

Answer

Number
of persons Rent
Controlled (X) Rent
Stabilized (Y) E(X) E(Y) E(X2) E(Y2)
1 0.61 0.41 0.61 0.41 0.61 0.41
2 0.27 0.3 0.54 0.6 1.08 1.2
3 0.07 0.14 0.21 0.42 0.63 1.26
4 0.04 0.11 0.16 0.44 0.64 1.76
5 0.01 0.03 0.05 0.15 0.25 0.75
6 0 0.01 0 0.06 0 0.36
Total 1 1 1.57 2.08 3.21 5.74

A. What is the expected value of the number of persons living in each type of unit?

E(X) =1.57
E(Y) = 2.08

B. What is the variance of the number of persons living in each type of unit?
Variance of X =E(X2)-E(X)2 = 3.21-(1.57)2 = 0.7451

Variance of Y =E(Y2)-E(Y)2 = 5.74-(2.08)2 = 1.4136

C. Make some comparisons between the number of persons living in rent-controlled units and the number of persons living in rent-stabilized units.

The Mean value is less in rent-controlled units and variability is also small. The mean value in rent-stabilized units is high with high variability

27. Consider a binomial experiment with n = 20 and p = .70

The probability density function is given by
, n = 20, p=0.70, x = 0,1,2,....20
The pdf is given ...

#### Solution Summary

The solution gives answers to problems related to binomial random variable. Step by step procedure to calculate probability, mean, variance, expectation and standard deviation are given.

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