Practice Test 2

Read and answer each question carefully. Show all work and give answers in blue books! Carry out all calculations and answers to 4 decimal places where applicable!

 

  1. At a checkout counter customers arrive at an average of 2.5 per minute. Find the probabilities that
    1. At most 3 will arrive in a given 1 minute interval.
    2. At least 3 will arrive in a given 1 minute interval.
    3. Exactly 14 will arrive during a given interval of 6 minutes.

 

  1. Acme Rental Cars has 10 Foreign cars and 15 American cars. If you choose 5 cars at random, find the probability that:
    1. Exactly 3 are Foreign.
    2. Less than 3 are Foreign.
    3. At least 3 are Foreign.

 

  1. You need to get to Detroit by tomorrow. You can book a coach flight now for $80. An airline offers a promotion to fly standby for $40. The probability of getting a standby flight is 0.40. If you miss the standby flight, you will need to fly first class the next day for $100. If your goal is minimize your expected cost, what should you do? Show why you made your choice.

 

  1. A new drug is developed to reduce blood pressure. Thirty randomly selected hypertensive patients (persons suffering from high blood pressure) receive the new drug. The manufacturer claims that the drug will work in 90% of patients.
    1. If manufacturer’s claim is true, what is the probability of observing blood pressure drops in 27 or more of 30 random patients?
    2. If manufacturer’s claim is true, how many of the 30 patients would you expect to see blood pressure drops for?
    3. If in a random sample of 30 patients, 26 experienced blood pressure drops, would you believe the manufacturer? Justify!
    4. In a random sample of 1000 patients, 850 experienced blood pressure drop,  would you believe the manufacturer? Justify!

 

  1. Farmer Abe has a roadside fruit and vegetable stand. His daily demand for tomatoes is Normally distributed with mean 30 pounds and standard deviation 8.
    1. If Abe has 23 pounds of tomatoes at the beginning of the day, what is the probability they will all be sold by days end?
    2. How many pounds of tomatoes must Abe have available at the beginning of the day so that there is only a 1.5% chance all the tomatoes will be sold that day?
    3. What is the probability Abe sells between 20 and 40 pounds of tomatoes on a given day?

 

  1. Professor Frink invents a new light bulb, whose lifetime follows an exponential distribution with a mean of 250 hours.
    1. What is the standard deviation of the lifetime of the light bulb?
    2. What is the probability the light bulb lasts more than 260 hours?
    3. What is the probability the light bulb lasts less than 10 DAYS?

 

Answers:

 

1.                  Y ~ Poisson (2.5)

a.       P(Y <= 3) = .7576

b.      P(Y >= 3)  = 1 – P(Y <= 2) = 1 - .5438 = .4562

c.       Y ~ Poisson ( 15), P(Y= 14) = .1024

 

2.                  Y ~ HyperGeometric ( N=25, n = 5, r = 10) = # of foreign cars.

a.       P( Y = 3) = .2372

b.      P(Y < 3) = P(Y <= 2) = .6988

c.       P(Y > = 3) = 1 - .6988 = .3012

 

3.                  E (Option 1 ) = 80 * 1 = $80

E (Option 2 ) = 40 * .4 + 100 * .6 = $76. so it is cheaper!

 

4.                  X ~ Bin (30, .9)

a.       P(X > = 27) = 1 - .3526 = .6474

b.      E ( X ) = 30 * .9 = 27

c.       P( X <= 26 | p = .9) = 1 = .6474 = .3526, yes because .3526 > .05

d.      Use normal approximation, npq = 90 > 5

P( X < = 850 ) = P( Z < -5.26) = 0. Do not believe the manufacturer!

 

5.                  X ~ N(30, 8)

a.       P( X > 23) = P( Z > -.88) = .8106

b.      P(Z < a ) = .9850, therefore a = 2.17 so x = 2.17 * 8 + 30 = 47.36

c.       P( 20 < X < 40)  = P(-1.25 < Z < 1.25) = 2*.3944 = .7888

 

6.                  X ~ Exp(250)

a.       Stdev = 250

b.      P(X > 260 ) = .3535

c.       P(X < 240) = 1 - .3829 = .6171