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(MATH244)midterm01F_L2.pdf
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Mid-term Examination
MATH 244 (L2) Applied Statistics
Mid-Term Examination October 26, 2001
Time allowed : 90 minutes Answer all questions.
1. (25 marks) The following data are the energy consumptions (in 100kWh) of 20 households in a specific month.
5.6 6.3 6.4 6.6 6.7 6.9 7.0 7.0 7.2 7.3
7.4 7.5 7.6 7.6 7.7 7.9 7.9 8.2 8.2 8.4
(a)
Construct a stem-and-leaf display for the data. (6 marks)

(b)
Is the distribution symmetric, positively skewed, or negatively skewed? (4 marks)

(c)
Find the five number summary. (10 marks)

(d)
Sketch a box-plot for the data. (5 marks)


2. (25 marks) The daily power demand in a moderate sized city follows a normal distribution. The expected daily power demand is 15 million kilowatt hours (MKWH), and the variance is 4 MKWH2. Suppose that the demands for power are independent from day to day.
(a)
What is the chance that the demand for a given day exceeds 17 MKWH? (5 marks)

(b)
A day with power demand exceeding 17 MKWH is considered to be a (7 marks) high power demand (HPD) day. What is the probability that there will be at least 30 HPD days in six months? (Assume 1 month = 30 days.)

(c)
Suppose that the local power corporation has an unlimited supply of (7 marks) power, and that revenues are based on $0.015 per kilowatt-hours. Find the probability that the TOTAL revenue for the power corporation in the next six months will be greater than 40 million dollars.

(d)
If the normal assumption is violated (i.e. the actual distribution of daily (6 marks) power does not follow a normal distribution), will the calculations in (a),


(b) and (c) still be valid? Explain briefly.
P. T. O.

P. 1
Mid-term Examination
3.
(10 marks) Three machines I, II and III manufacture 30%, 30% and 40%, respectively, of the total output of certain items. Of them, 4%, 3% and 2%, respective, are defective. One item is drawn at random, tested and found to be defective. What is the probability that the item was manufactured by (a) machines I, (b) machine II, (c) machine III?

4.
(10 marks) Suppose that the time for a student to finish a statistics examination is a gamma random variable with mean 90 minutes and standard deviation 15 minutes.

(a)
What are the values of the parameters and ? (5 marks)

(b)
How long should the examination last for so that 90% of the students can (5 marks) have enough time to complete the examination?



5.
(30 marks) The P&G company manufactures a shirt with two buttons on the front placket and one button on each sleeve cuff. Let X denote the number of buttons missing from the front placket and let Y denote the number of buttons missing from the sleeve cuffs of a randomly selected shirt. The joint probability mass function for X and Y is shown below:


Y
X 0 1 2
0 0.90 0.03 0.02
1 0.03 0.01 0.01

(a) Are X and Y independent? Why? (5 marks)
(b) Find ()E X , ()E Y , ()Var X , (