# Mathematics

1. A large industrial firm purchases several new word processors at the end of each year. The exact number of word processors purchased depends on the frequency of the number of repairs in the previous year. Suppose that the number of word processors x that are purchased each year has the following probability distribution (10 pts):

## x 0 1 2 3

f(x) 1/10 3/10 2/5 1/5 .

The cost of the desired model is $1200 per processor and a discount of 50 X^{2 } dollars is credited towards any purchase. There is also a fixed cost of $600 regardless of the number of processors purchased. How much can this firm expect to spend on new word processors?

2. The density function of a continuous random variable X, the total number of hours, in units of 100 hours, that a family runs a vacuum cleaner over a period of one year is described by the following probability density function: (20 pts)

x for 0< x < 1
2 – x for 1< x < 2
0 elsewhere |

í |

f(x) = |

a. Plot the PDF and CDF between the values of 0 < x < 2

b. What is the average (mean) number of hours per year that families run their vacuum cleaners

c. Find the 60^{th} percentile number of hours per year that families run their vacuum cleaners

d. Find the probability that families run their vacuum cleaners between 50 and 120 hours per year

3. Find the probabilities for the following discrete random variables (20 pts)

a. Among a shipment of 5,000 tires, 1,000 are slightly blemished. If one purchases 10 of these tires, what is the probability that 3 or less (P (x <= 3) are blemished?

b. The probability that a hockey game will go into overtime is .3; Of the next 12 hockey games, what is the probability that more than 3 (P(x >3)) will go into overtime?

c. The probability that a person, living in a certain city, owns a dog is estimated to be 0.3. Find the probability that the tenth person randomly interviewed in that city is the sixth one to own a dog.

d. A local drugstore owner knows that, on average, 100 people per hour stop by his store. Find the probability that in a given 3 minute period, more than 5 (P(x > 5) people enter the store.

e. An electronic switching device occasionally malfunctions and may need to be replaced. It is known that the device is satisfactory if it makes, on average no more than .2 errors per hour. A particular five hour period is chosen as a “test” on the device. If no more than 1 error occurs, the device is considered satisfactory. What is the probability that a satisfactory device will be mis- diagnosed as “unsatisfactory” on the basis of the test?

f. A restaurant chef prepares a tossed salad containing, on average, 5 vegetables per day. Find the probability that the salad contains more than 5 vegetables on i) a given day and on ii)3 of the next 4 days?

4. An investment project is expected to earn $100,000 on average with a standard deviation of $50,000. Assume earnings from this investment project is a normal distribution (10 pts)

a. Determine the probability that the annual net cash flow will be negative

b. What is the annual net cash flow at a probability of 75%

c. Of the next 15 people that invest in this project, what is the probability that 5 or less people will result an annual net cash flow of at least $100,000

d. What would the standard deviation have to be if the mean is 100,000 and the probability that a random variable x is less than 109,000 is 60%; (P(x< 109,000) ) = .60?

5. In a study designed to gauge married women’s participation in the workplace today, the data (provided in the excel template) are obtained from a sample of 750 randomly selected married women. Consider a woman selected at random from this sample in answering each of the following questions. (10 pts)

a. What is the probability that this randomly selected woman has a job outside the home?

b. What is the probability that this randomly selected woman has at least one child?

c. What is the probability that this randomly selected woman has a full time job and no more than one child?

d. What is the probability that this randomly selected woman has a part time job given she has one child?

6. A fast-food restaurant sells hamburgers and chicken sandwiches. On a typical weekday, the demand for hamburgers is normally distributed with a mean 313 and a standard deviation 57; the demand for chicken sandwiches is normally distributed with a mean 93 and a standard deviation 22. (12 pts)

a. How many hamburgers must the restaurant stock to be 98% sure of not running out of stock on a given day?

b. Answer part a for chicken sandwiches.

c. If the restaurant stocks 400 hamburgers and 150 chicken sandwiches for a given day, what is the probability that it will run out of hamburgers or chicken sandwiches (or both) that day? Assume that the demand for hamburgers and the demand for chicken sandwiches are independent

7. The average SAT test scores on both verbal and math components have been computed for students in 50 states and the District of Columbia. Also the proportion of high school graduates taking the test is indicated in the file. (18 pts) Refer to the SAT Scores Data provided in the excel template.

a. Calculate the mean, median and mode and standard deviation of the scores for verbal and math distributions

b. Based on the empirical data, what is the probability of scoring above 550 in math?

c. Using normal distribution theory, what is the probability of scoring above a 550 in math?