MSA 640 Homework Assignment #1 Due Friday, August 27, 2010 (100 Points Total/20 Points per Question)

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1 MSA 640 Homework Assignment #1 Due Friday, August 27, 2010 (100 Points Total/20 Points per Question) The numerical answers for most of these problems are provided. Consequently, grading will be based almost entirely on the correctness and clarity of the procedure and formulas you use to obtain your answer. You may use a calculator, laptop computer, or any tool you wish, but you must show all your work. You may consult with or discuss these problems with anyone you wish, but your work must be the product of your own personal effort. You may use additional paper if necessary. 1. An urn contains 12 balls described as follows: 5 are white (W) and lettered (L) 2 are white (W) and numbered (N) 4 are yellow (Y) and lettered (L) 1 is yellow (Y) and numbered (N) Find the following probabilities for one ball drawn from the urn: 5 balls white (W) and lettered (L) 2 balls white (W) and numbered (N) 4 balls yellow (Y) and lettered (L) 1 ball yellow (Y) and numbered (N) P(WL) = P(YL) = P(WN) = P(YN) = Suggestion: Complete the following table for use in computing the probabilities. L N W Y P(W) = P(L) = P(Y) = P(N) = N.B.: P(WL) = 0.42, P(WN) = 0.17, P(L) = 0.75, P(N) = 0.25 Homework #1 Page 1 of 6

2 #2 Last year at Northern Manufacturing Company, 550 people had colds during the year. There were 340 people who did no exercising and had colds, while all other people with colds were involved in a weekly exercise program. Sixty-eight percent (68%) of the 2,600 employees were involved in some type of exercise.use the following variables and show all formulas that you use: E = people who exercised, ~E = people who did not exercise, C = people who got colds Given: P(C), P(~E and C), P(E and C), P(E), P(~E) (a) What is the probability that an employee will have a cold next year? (0.21) (b) Given that an employee is involved in an exercise program, what is the probability that he or she will get a cold next year? (0.12) (c) Given that an employee is not involved in an exercise program, what is the probability that he or she will get a cold next year (0.41) (d) Explain whether exercising and getting a cold are independent events. Prove your answer using the conditional probability equation. Hint: If P(C E) is equal to P(C), then events C and E are independent, otherwise, they are dependent. Suggestion: Complete the table to the right for use in computing the probabilities. C ~C E ~E (a) P(C) = (b) P(C E) = (c) P(C ~E) = (d) Prove whether C and E are independent events. #3 The time to complete a construction project is normally distributed with a mean of weeks and a standard deviation of 3 weeks. In addition to producing numerical answers, complete drawing and labeling the following normal curves to show approximate solutions for each part of this problem. Show all the formulas you use. (a) What is the probability the project will be finished in fewer than 64 weeks? (0.75) (b) What is the probability the project will be finished in fewer than 59 weeks? (0.16) (c) What is the probability the project will take longer than 63 weeks? (0.37) (d) What is the probability the project will take between 58 and 66 weeks? (0.82) Homework #1 Page 2 of 6

3 #3 (continued) Given information: µ = σ = (a) P(X < 64) Show work in this space (b) P(X < 59) (c) P(X > 63) (d) P(58 < X < 66) Homework #1 Page 3 of 6

4 #4 Armstrong Faber produces a standard number two pencil called Ultra-Lite. Since Chuck Armstrong started Armstrong Faber, sales have grown steadily. With the increase in the price of wood products, however, Chuck has been forced to increase the price of the Ultra-Lite pencils. As a result, the demand for Ulta-Lite has been fairly stable over the past 6 years. On the average, Armstrong Faber has sold 550 thousand pencils each year. Furthermore, 90% of the time sales have been between 540 and 560 thousand pencils. It is expected that the sales follow a normal distribution with a mean of 550 thousand pencils. Given this information, find the standard deviation of this distribution. Complete drawing and labeling the following normal curve to show an approximate solution and show all intermediate work, including formulas. Hint: Start by working backward to find the Z value. (Z-value = 1.645, standard deviation = 6.08) 550 Use this space to show how the value of Z was obtained Use this space to show how the value of the standard deviation (σ) was obtained Homework #1 Page 4 of 6

5 #5 During the summer months in Washington, D.C., bus and subway ridership is believed to be heavily tied to the number of tourists visiting the city. During the past 12 years, the following data has been obtained where the X column represents millions of tourists and the Y column represents ridership in millions. (a) Develop the slope and intercept for the regression equation. (b 0 =0.539, b 1 =0.146) (b) What is the expected ridership if 20 million tourists visit the city? (3.46) (c) What is the expected ridership if 21 million tourists visit the city? (3.51) Don t forget to show all formulas that you use as well as all intermediate work. X Y ( X X ) ( Y Y ) ( X X ) 2 ( X X ) ( Y Y ) ( X ) ( ) 2 Y ( X ) ( Y ) ( X X ) ( X X ) ( Y Y ) X Y X = Y = b 1 = b 0 = The regression equation is: Forecast the value of Y for X = 20 using the regression equation: Forecast the value of Y for X = 21 using the regression equation: Homework #1 Page 5 of 6

6 Some Useful Formulas P( Aand P ( A = P( P ( Aand = P( A P( Z = X µ σ X µ σ = Z X = µ + ( Z σ ) x X = n y Y = n b ( X X ) ( Y Y ) 1 = 2 ( X X ) b0 = Y b1 X ˆ = b + b X Y 0 1 Homework #1 Page 6 of 6