Modeling and STAT

Question 1

When Does X have a binomial distribution with parameters n and p? Denoted as X~Bin(n,p)

Answer 1

If a total of n Bernoulli are conducted and
- trials are independent
- same probability of success
- x is a # success in the ‘n’ trials

Question 2

Binomial distribution requirements

Answer 2

Binary Outcomes
Independent Trials
N # of trials
Same prob per trial

Question 3

P(X≤2) when X~Bin(5,0.4)

Answer 3

Question 4

If X~Bin(n,p), p(x) = P(X = x) = …

Answer 4

Question 5

Bernoulli Probability

Answer 5

f(k;p) = pk + (1-p)(1-k)

Question 6

General Formula for Error propagation

Answer 6

y = (x-mu)/ sigma

Question 7

What does Mu represent?

Answer 7

The center of the distribution graph

Question 8

What does sigma represent?

Answer 8

plus or minus one standard deviation

Question 9

What percentage doe one standard deviation from the center of a normal distribution curve encompass?

Answer 9

68%

Question 10

What percentage does two standard deviations represent on a normal distribution curve?

Answer 10

95%

Question 11

What is the Z-score and Z-score table used for?

Answer 11

Will give number of exact standard deviations from center

Question 12

3 ways of finding normal distribution?

Answer 12

Generalized, standardized, and with the distribution table

Question 13

The lifetime of a lightbulb is normally distributed with a mean of µ = 1400 hours and a SD = 200

What is the probability that exactly two of them have lifetimes between 1350 and 1550 hours.

Answer 13

0.237

Question 14

Scores on an exam are normally distributed. 10% were below 60 and 80% were below 81.
Find the mean and standard deviation.

Answer 14

Mean. = 73.07
Std dev. = 9.42

Question 15

In a certain state, license plates consist of three letters followed by three numbers.
1. How many different license plates can be made?
2. How many different license plates can be made in which no letter or number appears more than once?

Answer 15

1. (26^3) (10^3) = 17,576,00
2. (26)(25)(24)(10)(9)(8) = 11,232,000

Question 16

Probability of an ‘Or’ function formula

Answer 16

(P(A) + P(B)) - P(A and B)

Question 17

How to find the Mean in a normal distribution?

Question 18

You will finish school in about 2 years at the age of
23. You plan to work till you are 63 then retire. If
you want to retire with $3 million, how much
should you put each month into a fund that has a
9% interest rate compounding monthly

Answer 18

$630.00

Question 19

Today $5000 is deposited in an account that earns
1% per month. Additional deposits are made at the
end of each month for the next 10 years. The
deposits starts at 100 and increase by $10 each
month. The amount of money in the account at the
end of 10 years is closest to?

Answer 19

F= 5000(F/P,1%, 120) +100(F/A, 1%, 120) + 10(P/A,1%,120)(F/P,1%,120)

P = 5000 + 100(P/A,1%,120) + 10(P/A,1%,120)

$150,000

Question 20

You took a loan of $200,000 to buy a house at a
fixed interest rate of 3% for 20 years. Your loan
compounds monthly. What is your monthly

Answer 20

$66,400

Question 21

A set of cash flows begins at $20,000 for the
first year with a decrease of $2000 each year until n = 10. With an interest rate of 7%, what is the equivalent
uniform annual cash flow?

Answer 21

$12,108

Question 22

Alice invests a fixed percentage of her salary at the end of each year. She decided to start with
5 percent of her starting salary. For the next 20 years she expects her salary to increase by
3.5% annually and her plan is to increase her investment by the same rate. How much will the
investment be worth after 20 years of investment if the interest rate is 8% per year and her
starting salary was $60,000?

Answer 22

$3,000

Question 23

A chemical engineering firm installed a chemical plant at a cost of $1.2 million. This plant
will be used for ten years and then sold for $250,000. The company uses a Minimum
Attractive Rate of Return (MARR) of 20% per year for all of its business operations. As a
student doing an internship with this company, you are asked to determine the minimum
revenue required each year to realize the expected recovery and return. What is your
answer?

Question 24

The firm you work for was presented with an opportunity to invest in a project. The
following are the data on the project:
The initial investment required $60,000,000

Salvage value after 10 years $1,200,000
Gross income expected from the project $23,000,000 / yr
Operating Costs:
Labor $3,500,000/yr
Materials licenses, insurance, etc. $1,500,000/yr
Fuel and other cost $1,750,000/yr
Maintenance Cost $ 750,000/yr
The project is expected to operate for ten years. If your management expects to make 25% on its
investment before taxes, would you recommend this project using the present worth analysis?

Question 25

Two stamping machines are under consideration for purchase by a metal recycling company.
The manual model will cost $43,000 to buy with a ten-year life. Its annual operating costs
will be $3,500 with expected annual savings of $8,000. A computer-controlled model will
cost $52,000 to buy and it will have a ten-year life. It will return a savings of $12,500 a year
with annual operating costs of $6,500 for labor and $2,500 for maintenance. The company’s
minimum attractive rate of return is 8%. Which of the models will the company select using
present worth analysis?

Question 26

An article presents a new method for timing traffic signals in heavily traveled intersections. The effectiveness of the new method was evaluated in a simulation study. In 50 simulations, the mean improvement in traffic flow in a particular intersection was 653.5 vehicles per hour, with a standard deviation of 311.7 vehicles per hour.

Find a 95% confidence interval for the improvement in traffic flow due to the new system. Round the answers to three decimal places.

Answer 26

Given :
Sample Size, n= 50
Sample Mean, $$\mathrm{\bar{x}}$$ =653.5
Population standard deviation, $$\sigma$$ = 311.7

NORM.S.INVEST

Question 27

Suppose X is a geometric random variable that counts the number of trials necessary to obtain the first successful trial. If the probability of failure for any given trial is 0.42, what is the probability that the first success will occur on the first or second trial?
A) (0.58)(0.42)^0 + (0.58)(0.42)^1
B) (0.42)(0.58)^0 + (0.42)(0.58)^1
C) (26 over 1)(0.58)^1(0.42)^25 + (26 over 2)(0.58)^2(0.42)^24
D) (26 over 1)(0.42)^1(0.58)^25 + (26 over 2)(0.42)^2(0.58)^24
E) (0.42)(0.58)^1 + (0.42)(0.58)^2

Answer 27

(A) The probability of failure on any trial is 0.42, therefore the probability of success on any trial is 1-0.42=0.58. Since x is geometric, the probability of the first success occurring on the first or second trial is P(X=1) + p(X=2) = 0.58(0.42)^0 + 0.58(0.42)^1.

Question 28

A coin is weighted so that the probability of heads if 0.64. On average how many coin flips will it take for tails to first appear?
A. 1.56
B. 2
C. 2.78
D. 6.4
E. This cannot be determined since the total number of flips is unknown

Answer 28

A. For a geometric random variable , the average time to the first success is 1/p = 1/0.64 = 1.56

Question 29

In a binomial experiment with 45 trials the probablility of more than 25 successes can be approximated by P(z>(25-27)/3.29). What is the probability of successes of a single trial of this experiment?
A. .07
B. .56
C. .6
D. .61
E. .79

Answer 29

C The binomial distribution can be approximated by the normal distribution with u= np and o= sqrt(np(1-p)) when n is large enough. In this case, P(z>(25-27)/3.29) = P(z>(x-u)/o)) If np= 27 and n=45, p=0.6

Question 30

The distribution of the binomial random variable that counts the number of successful trials in 70, each of which have 40% probability of failure, can be approximated by which of the following distributions?
A) A slightly skewed right distribution with a mean of 28 and a standard deviation of 16.8.
B) A slightly skewed right distribution with a mean of 42 and a standard deviation of 4.1.
C) An aproximately normal distribution with a mean of 28 and a standard deviation of 16.8.
D) An approxiamtely normal distribution with a mean of 42 and a standard deviation of 4.1.
E) The number of trials is not large enough to make a determination.

Answer 30

(D) If the probability of failure is 40% then the probability of success is 60%. Further, since np greater than or equal to 5, the distribution can be approximated by a normal distribution with u=np=42 and o=sqrt[np(1-p)] = 4.1

Question 31

An experiment consists of 31 independent trials. The probability that any one trial is a failure is 45%. Which is the best estimate of the probability that more than 20 trials will be failures?
A) 0.01
B) 0.02
C) 0.11
D) 0.89
E) 0.99

Answer 31

(A) The described situation is binomcdf with n=31 and p=1-0.45=0.55 . The probability of more than 20 failures os equivalent to 10 or fewer successes, which is P(X less than or equal to 10) symbolically. This is best found using the calaculator: binomcdf (31, 0.55, 10).

Question 32

The distribution of the binomial random variable that counts the number of successful trials in 65, each of which have a 30% probabiility of success can be approximated by which of the following distributions?
A) A slightly skewed left distribution with a mean of 65 and a standard deviation of 19.5.
B) Slightly skewed eft distribution with a mean of 19.5 and a standard deviation of 3.7.
C) An approximately normal distribution with a mean of 65 and a standard deviation of 19.5.
D) An approximately normal distribution with a mean of 19.5 and a standard deviation of 3.7.
E) The number of trials is not large enough to make a determination.

Answer 32

D. Since np > 5, the distribution can be approximated by a normal distribution with u= np = 19.5 and o = sqrt(np(1-p)) = 3.7