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This is a final exam for a university-level course in probability and statistics for engineers. It includes 4 short answer questions and 12 multiple choice questions, covering topics such as probability mass functions, hypothesis testing, linear regression, and normal distributions.
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Final Exam for MAT Probability and Statistics for Engineers.
Time : 3 hours Professor : M. Zarepour & G. Lamothe
Name :
Student Number :
Calculators are permitted. It is an open book exam. There are 4 short answer questions and 12 multiple choice ques- tions. The exam will be marked on a total of 28 points.
Submit your answers for the multiple choice questions in the follo- wing table.
Question Answer Question Answer 1 7 2 8 3 9 4 10 5 11
Short Answer Questions
[4] 1. Let X be a random variable with the probability mass function
f (x) = c(1 + |x − 4 |), for x = 3, 4 , 5
and 0 otherwise.
(a) Find the value for c. (b) Find P (X = 4|X ≥ 4). (c) Compute the expected value of X. (d) Compute P (X ≥ μ), where μ = E[X].
(a) Based on the above histogram and normal probability plot, would it appear reasonable to assume that the tar content is normally distributed? Discuss.
(b) Do we have sufficient evidence at α = 5% to conclude that the true mean tar content is larger than 14 mg? (c) Construct a 95% confidence interval for the mean tar content.
(Question 2 cont.)
(a) Below is a scatter plot of y against x. Does it appear to be reaso- nable to use the simple linear regression model y = β 0 + β 1 x + ε? Discuss.
(b) Write down the estimated regression line and use it to estimate the mean weight in grams for a period of x = 5 hours. (c) Compute a 95% confidence interval for the mean weight for a period of x = 5 hours.
(d) Find the coefficient of determination and interpret within the context of the problem.
(Question 3 cont.)
Multiple Choice Questions
Submit your answers for the multiple choice questions in the table found on the front page.
[1] 1. In a box of 12 light bulbs, there are 3 defective items. An inspector inspects 3 light bulbs selected at random and without replacement. Find the probability that there are exactly 2 defective light bulbs in his sample.
(A) 0.0791 (B) 0.0066 (C) 0.026 (D) 0.7 (E) 0.
[1] 2. It is known that the manufacturing time (in hours) of a certain pro- duct is normally distributed with mean μ and variance σ^2 = 0.25. What sample size is required so that we have 90% confidence that the maximum error of the estimate of μ is 0.05?
(A) 384 (B) 385 (C) 271 (D) 280 (E) 250
[1] 3. An operator receives on the average 20 calls per hour in accordance with a Poisson process. What is the probability that she waits more than 12 minutes before receiving the first call?
(A) 0.9816 (B) 0.9084 (C) 0.0916 (D) 0.0183 (E) 0.49.
[1] 4. If X and Y are two random variable such that
E(X) = E(Y ) = 0, E(X^2 ) = E(Y 2 ) = 1
and E((X − Y )^2 ) = 4. Then the correlation coefficient between X and Y is
(A) 0 (B) 0.5 (C) 1 (D) -0.5 (E) -
[1] 5. The probability that a machine produces a defective item is 0.01. Each item is checked as it is produced. Assume that these are independent trials. Compute the probability that at least 50 items must be checked to find one that is defective.
(A) 0.56 (B) 0.82 (C) 0.64 (D) 0.61 (E) 0.
[1] 6. If the cumulative distribution function of a random variable X is given by F (x) =
1 − (^) x^42 , for x > 2 0 , for x ≤ 2. Compute the probability that P (4 < X < 5).
(A) 0.09 (B) 0.91 (C) 0.25 (D) 0.75 (E) 0.
[1] 7. If the amount of cosmic radiation to which a person is exposed while flying by plane across Canada is a normal random variable with μ = 4 .35 mrem and σ = 0.59 mrem, find the probability that a person on such flights will be exposed to at least 5.50 mrem.
(A) 0.9744 (B) 0.2561 (C) 0.5000 (D) 0.3576 (E) 0.
[1] 8. Fifteen bearings made by a certain process have a mean diameter of 0.506 cm with a standard deviation of 0.004 cm. Compute the standard error of the estimate of the mean.
(A) 0.004 (B) 0.001 (C) 0.506 (D) 0.0003 (E) 0.
[1] 9. Refer to Question 8. Assume that the diameter of a bearing is normally distributed. Compute a 95% confidence interval for the mean diameter.
(A) [0.504,0.508] (B) [0.502,0.510] (C) [0.500,0.512] (D) [0.505,0.507] (E) [0.498,0.514]
[1] 10. Let X 1 ,... , X 20 be a random sample from a normal population with mean μ = 5 and variance σ^2 = 1.5. Let X be the sample mean. Find c such that P
σ/
< c
