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MATH 244 Applied Statistics
Final Examination December 17, 2001
Time allowed : 3 hours
Answer all six questions.

1. (22%) The height of soap bubbles in the dishpan is of importance to soap manufacturers. An experiment was performed by varying the amount of soap and measuring the height of the bubbles in a standard dishpan after a given amount of stirring. The data are as follows:
Grams of Product (X) 4.0 4.5 5.0 5.5 6.0 6.5 7.0
Bubble Height in mm (Y) 33 43 46 51 53 61 63

You are asked to analyse the data based on the following regression model. iii XY ++= , i = 1,2,...,7 ; ( 2 )0,~ N iid i
(a) (b) Fit the regression line of Y on X. Construct the ANOVA table. (5 marks) (6 marks)
(c) Calculate the coefficient of determination. (2 marks)
(d) Find the 95% prediction interval of the height of bubbles when 5 grams of soap product is used. (4 marks)
(e) Is the prediction of 45mm, based on the third pair of values, inferior to the prediction by the regression line obtained in (a)? (2 marks)
(f) An experimenter stated that the model ++= XY was a ridiculous model unless = 0 , for anyone knows that if you dont put any soap in the dishpan there will be no soap bubbles. Thus he insists on using the model += XY . Comment on the experimenters statement. (3 marks)

P. T. O.

P. 1
2. (18%) An amplifier circuit must be designed to achieve a gain of 100. The minimum
circuit, which will achieve this gain, is shown in figure (i) below. It consists of two small amplifiers; each small amplifier magnifies the volume of sound by a factor of 10. Two alternative circuits are shown in (ii) and (iii) which will achieve the desired gain. Each circuit can function as long as signals can pass from A to B. Suppose each individual small amplifier has a reliability (chance of functioning properly) of 0.9. Assume that they are functioning independently.




B
(ii) A



B
(iii) A
Amplifier 2
Amplifier 4
(a) Compute the reliability (chance of functioning properly) of each of the three circuits. Which circuit is the most reliable? (8 marks)
(b) Let X be the number of amplifiers functioning properly in circuit (iii). Write down the distribution, expected value and variance of X. (6 marks)
(c) If two of the amplifiers in circuit (iii) were broken down, what is the probability that this circuit is still functioning? (4 marks)

P. 2
3. (15%) The following data are guesses of the outcome of Super Bowl 1994 by 16 faculty
members and staff. Team = Buffalo (B) or Dallas (D); Points = total number of points to be scored in the game.
Staff Team Points
Alice B 46
Barbara D 56
Carole B 49
Dan D 66
Dave D 61
Emmy D 40
Erl B 33
Jack B 57

Staff Team Points
Jay D 45
Joan B 49
Joe D 68
Larry B 60
Lynne B 51
Ned D 45
Nick B 45
Ralph D 59

(a)
What are the