Download Fall 2009 Final Review

Fall 2009 Final Review Problems
1. The PDF for a continuous random variable is given below:
𝑓(𝑥) =
a.
b.
c.
d.
e.
f.
1
∗ (9 − 𝑥 2 ) 0 < 𝑥 < 3
18
0
𝑒𝑙𝑠𝑒
Show this is a valid PDF.
What is the CDF of this random variable?
What is the probability x=2.56?
What is the probability x is less than 1?
What is the probability x>2?
Without actually calculating a value “x”, write down the equations to find
the median and 90th percentile.
2. A random variable X is uniformly distributed from 10 to 30.
a. Find the PDF and CDF of X.
b. What is the probability x<20?
c. What is the probability x>25 given it is >15?
3. The cable guy will arrive to your apartment uniformly in the next 45 minutes.
a. What is the probability he arrives within 15 minutes?
b. What is his expected arrival time and standard deviation?
c. 20 other people are in the same situation, waiting for a cable guy to arrive
uniformly in the next 45 minutes. What is the probability exactly 6 of the 20
people have the cable arrive within 15 minutes?
4. A large multinational soda company, PeepCo, has a total of 230 subsidiaries
worldwide. The company’s accounting records show that the annual net income of its
subsidiaries follows a normal distribution with mean $560 and standard deviation of
$64 (both in thousand dollars).
a. A subsidiary of PeepCo reported a projected net income of no more than
$426 (thousand dollars) for the year 2009, what is the probability for that?
b. Also, about 38% of its subsidiaries reported a net income between $460 and
$580 (thousand dollars) for the year 2008, is this higher or lower than
expected? Why?
c. The company headquarters usually gives a bonus to the top 15% of its
subsidiary managers, (assuming there is one manager in each subsidiary). If
the company uses its accounting records to decide the cutoff, how much net
annual income must a subsidiary generate to be qualified for a bonus?
5. Elmo is selling bottled lemonade at a foodstand. He wants to know the average
amount of lemonade in his bottles. His friend Ernie checked 25 bottles and found the
average amount is 11.5 ounces. He was told each bottle has a standard deviation of
2.25 ounces. Assume the amount of the lemonade follows a normal distribution and
answer the following questions.
a. If Ernie’s measure is correct, construct a 98% confidence interval for the sample
average of 25 bottles and help Ernie tell what he has found to Elmo.
b. Elmo is worried about his inventory; he wants to check whether his lemonade
really has an average of 12 ounces. His tolerable margin of error is 0.4 ounces
(with 95% confidence). He knows from the manufacturer that his bottles have a
standard deviation of 1.25 ounces, please help him decide how many bottles he
has to sample to get a tolerable margin of error.
6. The lifetime of a certain brand of automobile tire is exponentially distributed with a
mean of 4 years.
a. Find the probability that a randomly selected tire lasts between 3.5 and 4.5
years.
b. If a randomly selected tire lasts more than 3 years, what is the probability is lasts
more than 4.5 years?
c. What is the median lifetime of a randomly selected tire?
7. A hotel has 100 rooms and the probability a room is occupied on any given night is 0.6.
The probability of occupancy would actually depend on many factors such as the
season, but for simplicity we assume the overall occupancy rate of 60% only depends on
external factors. Let X represents the number of rooms occupied on a random selected
day.
a. What is the exact distribution of X and its parameter(s)?
b. Write an expression for the probability that 60 to 70 rooms are occupied.
c. What distribution is a good approximation to X, and its parameter(s)?
d. What is the approximate probability that 60 to 70 rooms are occupied,
inclusive?
8. A sketch artist will either draw in black and white or use colored pencils. The
amount of time it takes the artist to complete a black and white sketch is normally
distributed with a mean of 20 minutes and a standard deviation of 6 minutes. The
amount of time it takes the artist to complete a colored pencil sketch is normally
distributed with a mean of 44 and a standard deviation of 12. Let W be the length of
time it takes the artist to complete a black and white sketch and C be the length of
time it takes the artist to complete a sketch using colored pencils.
a. What is the 90th percentile of C?
b. What is the median time it will take the artist to finish a black and white
sketch?
c. What is the probability that he will finish a sketch using colored pencils
(C) in less than the median time it takes him to finish a black and white
sketch (W)?
9. The length of a novel has a mean of 260 pages and a standard deviation of 30 pages.
A local library has 40 books they are trying to put on an empty shelf, but the shelf
will only hold a maximum of 10,000 pages total. What is the probability that all 40
books will be able to fit on the shelf? Name the distribution and parameters you are
using.
10. The final scores of students in a large chemistry class are approximately normally
distributed with a mean of 72 points and a standard deviation of 8.
a. How many students in a section of 100 do you expect to end up with a score
below 50?
b. Where is the highest the C cutoff could be made and still have at least 75% of
the students get a C or better?
11. A set of fair die is tossed repeatedly 360 times. Let X be the number of times
double-sixes are tossed. By using the Normal approximation to the Binomial,
compute
a. 𝑃(8 ≤ 𝑋 ≤ 12)
b. 𝑃(8 < 𝑋 < 12)
12. A game consists of drawing two cards from a standard deck of 52 cards without
replacement. For every face card you win $10 and for an ace you win $20. For any
other card you lose $1. Let X be the amount of money you win.
a. Find the PMF table for X
b. Compute the expected value and variance of your winnings
c. If you play this game repeatedly 40 times (the cards are put back after each
game) what is an approximate distribution of your total winnings?
13. Suppose you are simulating a large number of die rolls. How many simulations
would you have to run to be 90% sure your proportion of even rolls is between 48%
and 52%? (You can also consider this to mean the Margin of Error is at most .02)
14. A continuous random variable X has PDF
3
f(x) = 𝑘𝑥 (𝑥 − 2) 2 ≤ 𝑥 ≤ 4
0
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
a.
b.
c.
d.
e.
Find the value of k that makes this a valid PDF.
Compute the CDF.
Find the mean of this distribution.
Compute the probability that X is between 2.5 and 3.
Compute the probability that X is greater than 3.5
15. A continuous random variable X has the following CDF
0
𝑥 ≤0
F(x) = 𝑥 3 /8 0 < 𝑥 < 2
1
𝑥≥2
a.
b.
c.
d.
Find the PDF of X.
Find the mean and standard deviation of this distribution.
Compute the probability that X is between 0.2 and 0.5.
Suppose we take two independent observations from this distribution. What is
the probability that both are greater than 0.60?
16. On a game show contestants play a game by putting minigolf balls from a five yard
distance. Every contestant gets three chances to make a put. Assume that every
contestant has an equal chance of 0.2 of sinking a ball. The game is won, if at least
one of the three balls can be sunk.
a. What is the probability that a contestant wins this game? Which distribution
are you using and what are the parameters?
b. If 300 contestants line up to play this game, what is the approximate
probability that at least 150 of them win?
17. The amount of water in a 12-oz water bottle follows a normal distribution with
unknown mean µ oz and standard deviation of 0.08. We take a random sample of
64 water bottles and get a sample mean of 12.04 oz.
a. What is the distribution of the sample mean? What are you using to find this
distribution?
b. Find a 95% confidence interval for the average amount of water in a bottle.
18. A survey of 611 office workers investigated telephone answering practices, including
how often the office worker was able to answer incoming telephone calls and how
often incoming telephone calls went directly to voice mail. A total of 281 office
workers indicated that they never need voice mail and are able to take every
telephone call.
a. What are the sampling distribution and parameters for the proportion of
office workers that can take every telephone call?
b. What is the probability the proportion is below .44?
c. Construct a 90% confidence interval for the proportion of the population of
office workers who are able to take every telephone call.