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# Probability, Simulation, Demand, Sample Space, Decision

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18. Employees of a local company are classified according to gender and job type. The following table summarizes the number of people in each job category.

Male (M) Female (F)
Job
Salaried staff (SS) 30 50
Hourly staff (HS) 60 40

If an employee is selected at random, what is the probability that the employee is female given that the employee is a salaried staff member.
a) .1667
b) .5
c) .6
d) .625
e) .7

19. In the set of all past due accounts, let the event A mean the account is between 31 and 60 days past due and the event B mean the account is that of a new customer. The complement of A is
a) all new customers.
b) all accounts fewer than 31 or more than 60 days past due.
c) all accounts from new customers and all accounts that are from 31 to 60 days past due.
d) all new customers whose accounts are between 31 and 60 days past due.

41. The number of cars arriving at Joe Kelly's oil change and tune-up place during the last 200 hours of operation is observed to be the following:

Number of cars arriving Frequency
3 or less 0
4 10
0 30
6 70
7 50
8 40
9 or more 0
Based on the above frequencies, use two digit random numbers, start with random number 00 and determine the random number ranges for the data set given above.

39. A researcher wants to simulate sunny and rainy days in her town for a 3-week period. What is the minimum number of digits the student must obtain from a random number table for each observation if it rained on two-fifths of the days over the past several years at this time of the year? Assume that days can be classified historically as either sunny or rainy.

a) 1
b) 2
c) 3
d) all of the above.

42. The relationship d = 5000 - 25p describes what happens to demand (d) as price (p) varies. Here, price can vary between \$10 and \$50.

a. How many units can be sold at the \$10 price? How many can be sold at the \$50 price?
b. Model the expression for total revenue.
c. Consider prices of \$20, \$30, and \$40.
Which price alternative will maximize total revenue?
What are the values for demand and revenue at this price?

43. A package of candy contains 12 brown, 5 red, and 8 green candies. You grab three pieces from the package. Give the sample space of colors you could get. Order is not important.

44. There are two more assignments in a class before its end, and if you get an A on at least one of them, you will get an A for the semester. Your subjective assessment of your performance is
Event Probability
A on paper and A on exam .25
A on paper only .10
A on exam only .30
A on neither .35

a. What is the probability of getting an A on the paper?
b. What is the probability of getting an A on the exam?
c. What is the probability of getting an A in the course?
d. Are the grades on the assignments independent?

45. The high school GPA of applicants for admission to a college program are recorded and relative frequencies are calculated for the categories.
GPA F(x)
x < 2.0 .08
2.0 = x < 2.5 .12
2.5 = x < 3.0 .35
3.0 = x < 3.5 .30
3.5 = x

a. Complete the table to make this a valid probability distribution.
b. What is the probability an applicant's GPA will be below 3.0?
c. What is the probability an applicant's GPA will be 2.5 or above?

46. A calculus instructor uses computer aided instruction and allows students to take the midterm exam as many times as needed until a passing grade is obtained. Following is a record of the number of students in a class of 20 who took the test each number of times.
Students Number of tests
10 1
7 2
2 3
1 4

a. Use the relative frequency approach to construct a probability distribution and show that it satisfies the required condition.
b. Find the expected value of the number of tests taken.
c. Compute the variance.
d. Compute the standard deviation.

47. Lakewood Fashions must decide how many lots of assorted ski wear to order for its three stores. Information on pricing, sales, and inventory costs has led to the following payoff table, in thousands.
Demand
Order Size Low Medium High
1 lot 12 15 15
2 lots 9 25 35
3 lots 6 35 60

a. What decision should be made by the optimist?
b. What decision should be made by the conservative?
c. What decision should be made using minimax regret?

48. The table shows both prospective profits and losses for a company, depending on what decision is made and what state of nature occurs. Use the information to determine what the company should do.
s1 s2 s3
d1 30 80 -30
d2 100 30 -40
d3 -80 -10 120
d4 20 20 20

a. if an optimistic strategy is used.
b. if a conservative strategy is used.
c. if minimax regret is the strategy.

49. The drying rate in an industrial process is dependent on many factors and varies according to the following distribution.
Minutes Relative Frequency
3 0.22
4 0.36
5 0.28
6 0.10
7 0.04
Using these random numbers, simulate the drying time for 5 processes: 0.53; 0.95; 0.97; 0.96; and 0.07.

50. Use a four period moving average to forecast attendance at baseball games. Historical records show the attendance at ten consecutive baseball games as 5346, 7812, 6513, 5783, 5982, 6519, 6283, 5577, 6712, 7345.

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#### Solution Summary

Questions on Probability, Simulation, Demand function, Sample Space, Decision have been answerd.

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## Statistics 33 multiple choice: probability, distributions, decision, expected value, sample

1. Probability Type (10.0 points)
A ___________ probability is the probability that an event occurs given that another event has already occurred.

a) joint
b) marginal
c) conditional
d) exclusive

2. Probability of Multiple Events (10.0 points)
Jim is considering pursuing an MS in Information Systems degree. He has applied to two different universities. The acceptance rate for applicants with similar qualifications is 20% for University X and 45% for University Y. What is the probability that Jim will be accepted at both universities?

a) .65
b) .25
c) .20
d) .09

3. Binomial Probability (10.0 points)
A fair die is rolled nine times. What is the probability that an odd number (1,3 or 5) will occur less than 3 times?

a) .0899
b) .2544
c) .7456
d) .9101

4. Normal Probability Distribution (10.0 points)
A loaf of bread is normally distributed with a mean of 22 oz and a standard deviation of 0.5 oz. What is the probability that a loaf is less than 22.25 oz?

a) .1915
b) .3085
c) .6915
d) .7257

5. Normal Probability Distribution II (10.0 points)
The metropolitan airport commission is considering the establishment of limitations on noise pollution around a local airport. At the present time, the noise level per jet takeoff in one neighborhood near the airport is approximately normally distributed with a mean of 100 decibels and a standard deviation of 3 decibels. What is the probability that a randomly selected jet will generate a noise level of more than 105 decibels?

a) 0.0228
b) 0.0475
c) 0.0500
d) 0.0512

6. Joint Probabilities (10.0 points)
Employees of a local company are classified according to gender and job type. The following joint frequency table summarizes the number of people in each job category.

If an employee is selected at random, what is the probability that the employee is male and salaried staff?

a) 0.015
b) 0.100
c) 0.167
d) 0.267

7. Find the Conditional Probability (10.0 points)
Employees of a local company are classified according to gender and job type. The following table summarizes the number of people in each job category.

If an employee is selected at random, what is the probability that the employee is female given that the employee is a salaried staff member.

a) .1667
b) .50
c) .60
d) .625

8. Decision Making without Probabilities (10.0 points)
The local operations manager for the IRS must decide whether to hire 1, 2, or 3 temporary workers. He estimates that net revenues will vary with how well taxpayers comply with the new tax code.

If he uses the maximin criterion, how many new workers will he hire?

a) 1
b) 2
c) 3

9. Expected Value of Perfect Information EVPI (10.0 points)
The quality control manager for ENTA Inc. must decide whether to accept (a1), further analyze (a2) or reject (a3) a lot of incoming material. Assume the following payoff table is available. Historical data indicates that there is 30% chance that the lot is poor quality (s1), 50 % chance that the lot is fair quality (s2) and 20% chance that the lot is good quality (s3).

What is the maximum amount that you would be willing to pay for perfect information?

a) 20
b) 39
c) 57
d) 77

10. Decision Tree Expected Value (10.0 points)
Consider the following decision tree.

What is the expected value at node 4?

a) 600
b) 1600
c) 1800
d) 2500

11. Decision Tree Expected Value II (10.0 points)
Use the decision tree from question 10.

What is the value associated with node 3?

a) 600
b) 1600
c) 1800
d) 2500

12. Expected Value of Perfect Information II (10.0 points)
A business owner is trying to decide whether to buy, rent, or lease office space and has constructed the following payoff table based on whether business is brisk or slow.

If the probability of brisk business is .40 and for slow business is .60, determine the expected value of perfect information is:

a) 12
b) 55
c) 57
d) 69

13. Fill-in-the-blank (10.0 points)
The __________ of sample information is the difference between the expected value with and without additional information.

a) efficiency
b) utilization
c) expected value
d) events

14. Fill-in-the-blank II (10.0 points)
The __________ of sample information is the ratio of the expected value of sample information to the expected value of perfect information.

a) utilization
b) expected value
c) efficiency
d) events

15. Find the Simulation Value (10.0 points)

If a simulation begins with the first random number, what would the first simulation value be? (Hint: The random numbers are chosen from 0 - 100 with uniform distribution, so divide each random number by 100.)

a) 1
b) 2
c) 3
d) 4

16. Find the Simulation Value II (10.0 points)
Given this frequency distribution, the random number 0.61 would be interpreted as a demand of:

a) 0
b) 1
c) 2
d) 3

17. Complete the Simulation (110.0 points)
A graduate research assistant "moonlights" at the short order counter in the student union snack bar in the evenings. He is considering asking for help taking orders, but needs to convince the management that they should hire another student. Because he is taking a simulation class, he thinks it may be the perfect way to convince management to hire more help if he can show that customers have to wait a long time. When a customer arrives, he takes their order and their payment, prepares the food, gives it to the customer, and then takes the order from the next person in line. If someone arrives while he's cooking an order, they have to wait until he's completed the current order. He has simulated 5 orders.

Average customer waiting time is:

a) 1 minute
b) 2 minutes
c) 2.5 minutes
d) 3.0 minutes

18. Fill-in-the-blank II 2 (10.0 points)
The __________ method is the most common type of forecasting method for the long-term strategic planning process.

a) time series
b) regression
c) qualitative

19. Computer the Moving Average (10.0 points)
Given the following data on the number of pints of ice cream sold at a local ice cream store for a 6-period time frame:

You are at the beginning of period 6. Compute a 3-period moving average forecast for this period.

a) 246.67
b) 247.50
c) 283.33
d) 300

20. Compute 5-month Moving Average (10.0 points)
Given the following data on the number of pints of ice cream sold at a local ice cream store for a 6-period time frame:

You are at the beginning of period 6. Computer a 5-month moving average forecast for this period.

a) 237.0
b) 247.5
c) 257.0
d) 300.0

21. Find the Average Forecast Error (10.0 points)
The following data summarizes the historical demand for a product

Determine the average forecast error when using the 4-month moving average method.

a) 5.25
b) 5.75
c) 6.25
d) 6.75

22. Find the weighted average (10.0 points)
The following data summarizes the historical demand for a product

Use a weighted moving average method with weights w1 = .2, w2 = .3 and w3 = .5 and determine the forecasted demand for September.
HINT: Remember that the first weight refers to the most recent month.

a) 33.5
b) 35.5
c) 37.5
d) 38.5

23. Forecast Using Linear Regression (10.0 points)
Robert wants to know if there is a relation between money spent on gambling and winnings:

If he spends \$20, how much can he expect to win if he uses regression analysis?

a) 320
b) 370
c) 410
d) 450

24. Forecast Pattern Type (10.0 points)
__________ is an up-and-down repetitive movement within a trend occurring periodically.

a) Prediction
b) Seasonal pattern
c) Trend
d) Cycle

25. Choose the Correct Method Type (10.0 points)
__________ attempt to develop a mathematical relationship between the item being forecast and factors that cause it to behave the way it does.

a) Qualitative methods
b) Regression
c) Time series
d) Quantitative methods

26. Determine the graph trend type (10.0 points)
Consider the following graph of sales.

Which of the following characteristics is exhibited by the data?

a) Cyclical only
b) Trend plus cyclical
c) Trend plus seasonal
d) Seasonal only

27. Determine Type of Averaging Method (10.0 points)
__________ is an averaging method that reacts more strongly to recent changes in demand.

a) Weighted moving average
b) Exponential smoothing
c) Linear Trend Line
d) Moving average

28. Simple Exponential Smoothing (10.0 points)
Given an actual demand of 59, a previous forecast of 64, and an alpha of .3, what would the forecast for the next period be using simple exponential smoothing?

a) 57.5
b) 60.5
c) 62.5
d) 65.6

29. Determine the Regression Equation (10.0 points)
Consider the following annual sales data for 1996-2003, use the linear regression method and determine the estimated sales equation. HINT: Remember to use period numbers. Refer to your book's example on the linear trend line.

a) y = 2.63 + 0.21x
b) y = 2.63 - 0.21x
c) y = -0.21 + 2.63x
d) y = 0.21 - 2.63x

30. Cause-Effect Relationship (10.0 points)
Consider the following annual sales data for 1996-2003. This is the same table shown in the previous question. Determine the percentage of in the variation of sales that can be attributed to the year.

a) 91.54%
b) 93.22%
c) 95.34%
d) 96.55%

31. Extra Credit I - Normal Distribution (110.0 points)
The law firm of Dewey, Cheetham, and Howe has a pool of candidates wishing for internship in the law firm. The law firm has decided to test each candidate on his or her ability to win a nuisance lawsuit. Candidates who score in the top 33% will be given an internship. The law firm has issued similar tests in the past and knows that the average score is 80 (out of 100), the standard deviation is 10, and that test scores are normally distributed. What is the minimum score that a candidate must earn in order to place in the top 33%?HINT: In a normal situation, you would be given the x-value and would determine the area under the normal curve. Here, you are given the area under the normal curve and must determine the x-value.

a) 81.25
b) 84.40
c) 86.30
d) 88.60

32. Extra Credit II - Bayesian Analysis (110.0 points)
An airline is trying to determine whether the price of fuel will rise or fall in the near future in order to determine whether to negotiate a fuel contract now or later. The airline executives believe there is an 65% chance that the cost of fuel will fall. However, they are risk-averse and would prefer to hire an analyst to obtain more information. The airline executives have found that when the price of fuel fell, the analyst correctly predicted the decrease in cost 90% of the time. When the price of fuel rose, the analyst correctly predicted the increase 75% of the time. The executives hired the analyst, and the analyst issued a report predicting that the cost of fuel would fall. What is the probability that the price of fuel will fall given the analyst prediction of such a decrease? You should carry your calculations to 4 decimal places and round of your answer to 2.

a) .65
b) .68
c) .74
d) .87

33. Extra Credit III - Forecast Errors (10.0 points)
This question requires some thought, but the correct answer can be reached if you think long enough about it.You are performing an analysis of your chosen forecasting method and have tracked errors for forecasts over the previous 24 time periods. You have obtained the following results from your error analysis:MAD = 25
MAPD = 0.10