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BUSINESS STATISTICS

Dr. Azar Abizada

HOMEWORK 3

Due Date: Tuesday, October 28thclass time

True False questions (10 points: 2 points each).

1. The number of defective laptops in a store is a discrete variable while the life of alaptop battery is a continuous variable.

2. Conditional probability of event A given some other event B cannot be greater thanthe individual probability of event A.

3. Suppose Dr. Valiyev takes 7 penalty shots. Each shot is independent of the other.The probability that he will score a goal from a penalty shot is 0.2.Statement: The probability that he will score 4 penalty shots is more than the

probability that he will be score 3 penalty shots.

4. The correlation coefficient is always positive.

5. If two random variables X and Y are independent, then

P(y | x) = P(x) and P(x | y) = P(y).

Problem 1 (16 points):When a machine is properly calibrated, 0.5% of its output is

defective, but when it is out of adjustment, 6% of the output is defective. The

technician in charge of calibration can successfully calibrate the machine in 90% of

his attempts. The technician has just made an attempt to calibrate the machine.

a) (8 points) The first unit produced after the calibration effort is found to bedefective. Given this information, what is the probability that the machine is inadjustment?

b) (8 points) The first unit produced after the calibration effort is found to be

normal (not defective). Given this information, what is the probability that themachine is out of adjustment?

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Problem 2 (26 points): Be careful, each part of this question is independent of

one another.

a) (5 points)Suppose that the probability is 0.2 that the value of the U.S. dollar

will rise against the Chinese yuan over any given week and that the outcome

in one week is independent of that in any other week. What is the probability

that the value of U.S. dollar will rise against Chinese yuan at least twice over a

period of 7 weeks?

b) (5 points)A customer service center in India receives, on average 5.4

telephone calls per minute. If the distribution of calls is Poisson, what is the

probability of receiving at least 3 calls during next two minutes?

c) (5 points)X and Y are random variables with finite variance. Which one is

larger: Var(X+Y) or Var(X-Y)?

d) (5 points)Assume that each birth is an independent event with an equalchance of being male or female. If you have four children, they may all be ofone sex, there may be three of one sex and one of the other sex, or there may

be two of each. Which is most likely?

e)

(6 points)Explain, how do median, mean, standard deviation and range of thepopulation change if we multiply every observation in the data by a constant M?

Problem 3 (10 points): In its Sunday business supplement a local newspaper reported

that 33% of those employed full time work less than 40 hours per week, and 20.05%

of those employed work more than 49 hours per week. Assuming that the hours

worked per week by a randomly chosen full time worker in a randomly chosen week

is normally distributed find the population mean and standard deviation of that

distribution.

Problem 4 (15 points): Annual sales revenue for Meadow Flower Cereal is

$500,000.000. Jeffery Shower, a 52 year old senior vice president, is in charge of

production and sales of Nutty 8 cereal. Daily production in cases is normally

distributed with a mean of 100 and a variance of 625. Daily sales in cases are also

normally distributed with a mean of 100 and standard deviation of 8. Sales and

production have a correlation of 0.60. The selling price per case is $10. The variable

production cost per case is $7. The fixed production costs per day are $250. What is

the probability that total revenue is greater than total costs on any day?

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Problem 5 (10 points):Staff Inc., a management consulting company, is surveying

the personnel of Acme Ltd. It determined that 35% of the analysts have an MBA and

that 40% of all analysts are over age 35. Further, of those who have an MBA, 30% are

over age 35.

a)

(5 points)What is the probability that a randomly chosen analyst both has anMBA and also is over age 35?

b) (5 points)What is the probability that a randomly chosen analyst who is overage 35 does not have an MBA?

Problem 6 (12 points):There are two locations in town (north and south) underconsideration for a new restaurant, but only one location will actually becomeavailable. If it is in the north, the restaurant stands a 90% chance of surviving its firstyear. However, if it is built in the south, its chances of survival are only 65%. It isestimated that the chances of the northern location being available are 40%.

a) (6 points)Find the probability that the restaurant will survive its first year.

b) (6 points)Find the probability that the restaurant is in the south given that it is

successful.

Problem 7 (11 points): Boxes are filled with sugar by a machine that is set to deliver

an average of 20.3 ounces. The weights are normally distributed with a standard

deviation of 0.3 ounces.

a) (5 points) If a consumer advocate buys a package of the product and thenproceeds to weigh the contents, what is the probability that the package shehas purchased will have a content weight between 20 and 21 ounces?

b) (6 points) If a consumer advocate buys a package of the product and thenproceeds to weigh the contents, what is the probability that the package shehas purchased will have a content weight either above 21.2 or below 19.6ounces?