CHE 331 Engineering Statistical Design

Answers For Midterm Exam - Winter 2010

Question 1 (2-166) [25]

Natural red hair consists of two genes. People with red hair have two dominant genes, two regressive genes, or one dominant and one regressive gene. A group of 1000 people was categorized as follows:

Gene 2

Gene 1 Dominant Regressive Other

Dominant 5 25 30

Regressive 7 63 35

Other 20 15 800 Let A denote the event that a person has a dominant red-hair gene and let B denote the event that a person has a regressive red hair gene. If a person is selected at random from this group, compute the following.

A) P (A) B) P (A ∩ B) C) P (A B) D) P (A′ ∩ B) E) P (A|B)

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Question 2(Devores) [25] A certain sports car comes equipped with either an automatic or a manual transmission, and the car is available in one of four colors. Relevant probabilities for various combinations of transmission type and color are given in the accompanying table.

Color Transmission

Type White Blue Black Red

A 0.13 0.10 0.11 0.11 M 0.15 0.07 0.15 0.18

Let A = (automatic transmission), B = {black}, and C = {white}.

a. Calculate P(A), P(B), and P(A∩B). b. Calculate both P(A|B) and P(B|A), and explain in context what each of these

probabilities represents. c. Calculate and interpret P(A|C) and P(A/C′ ).

ANSWER: a. P(A) = .13 + .10 + .11 + .11 =.45,

P(B) = .11 + .15 = .26 P(A∩B) = .11

b. P(A|B) = ( ) .11 .4231( ) .26

P A BP B∩

= =

Knowing that the car is black, the probability that it has an automatic transmission is .4231.

P(B|A) = ( ) .11 .2444( ) .45

P A BP A∩

= =

Knowing that the car has an automatic transmission, the probability that it is black is .2444.

c. P(A|C) = ( ) .13 .4643( ) .28

P A CP C∩

= =

The probability that the car has automatic transmission, knowing that the car is white is .4643.

( | )P A C′ = ( ) .32 .4444( ) .72

P A CP C

′∩= =

′

Knowing that the car is not white, the probability that it has an automatic transmission is .4444.

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Question 3 (3-131) [25] The probability that your call to a service line is answered in less than 30 seconds is 0.75. Assume that your calls are independent.

a) If you call 10 times, what is the probability that exactly 9 of your calls are answered within 30 seconds?

b) If you call 20 times, what is the probability that at least 16 calls are answered in less than 30 seconds?

c) If you call 20 times, what is the mean number of calls that are answered in less than 30 seconds?

Solution:

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Question 4 [25] If X is a normal random variable with mean 85 and standard deviation 10, compute the

following probabilities by standardizing.

a. ( 100)P X ≤ b. ( 80)P X ≤ c. (65 100)P X≤ ≤ d. ( 70)P X ≥ e. (85 95)P X≤ ≤ f. ( 80 10)P X − ≤ ANSWER: a. ( 100) ( 1.5) (1.50) .9332P X P Z≤ = ≤ = Φ = b. ( 80) ( .5) ( .5) .3085P X P Z≤ = ≤ − = Φ − = c. (65 100) ( 2 1.5)P X P Z≤ ≤ = − ≤ ≤ (1.5) ( 2.0) .9332 .0228 .9104= Φ −Φ − = − = d. ( 70) ( 1.50) 1 ( 1.50) .9332P X P Z≥ = ≥ − = −Φ − = e. (85 95) (0 1.0) (1.0) (0) .3413P X P Z≤ ≤ = ≤ ≤ = Φ −Φ = f. ( 80 10) ( 10 80 10) (70 90)P X P X P X− ≤ = − ≤ − ≤ = ≤ ≤ = ( 1.5 .5) (.5) ( 1.5)P Z− ≤ ≤ = Φ −Φ − = .6915 - .0668 = .6247

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Question 5 [20] The length of a time in seconds that a user views a page on a Web site before moving to

another page is lognormal random variable with parameters θ = 0.5 and ω = 1.

(a) What is the probability that a page is viewed for more than 10 seconds?

(b) By what length of time have 50% of the users moved to another page?

(c) What is the mean and standard deviation of the time until a user moves from the

page?