HOMEWORK: EXPECTED VALUE AND SAMPLING DISTRIBUTIONS
EXPECTED VALUES:
PROBLEM 1: Expected Value
What is the expected value of a lottery ticket where there is only two chances in a million of winning
the grand prize of $10 Million?
PROBLEM 2: Expected Value
You have been offered a chance to purchase a lottery ticket with a 1% chance of making $1,000; 4%
chance of making $100; and 95% chance of making $0. The price of the ticket is $15. Should you
buy it? Why?
PROBLEM 3: Expected Value
You have been offered a business deal. You estimate that there is a 1% chance of making $100,000;
4% chance of making $40,000; a 20% chance of making $10,000; and 75% chance of making $0.
How much should you be willing to pay for this deal? Do you think you would actually pay that
much? Why?
SAMPLING DISTRIBUTIONS
PROBLEM 1:
Suppose that New York State high school average scores, for students who graduate, are normally
distributed with a population mean of 70 and a population standard deviation of 13.
a) The “middle” 95% of all NYS high school students have average scores between ______ and
________ ?
b) What proportion of NYS high school students have average scores between 60 and 75?
c) Calculate the 14th percentile.
d) Calculate the 92nd percentile.
PROBLEM 2 :
Suppose CUNY professors have an average life, normally distributed, of 80 years with a population
standard deviation of 9 years.
a)What percent of CUNY professors will live more than 96 years?
b) What percent of CUNY professors will not make it past the age of 60?
c) Calculate the 96th percentile.
d) Calculate the 2nd percentile.
e) What proportion of CUNY professors will live between 70 and 85 years?
PROBLEM 3:
Suppose the lifetimes of Hoover vacuum cleaners are normally distributed with an average life (μ) of
12 years and a population standard deviation (σ) of 1.4 years.
a) What proportion of Hoover vacuum cleaners will last 14 years or more?
b) What proportion of Hoover vacuum cleaners will last 9 years or less?
c) What proportion of Hoover vacuum cleaners will last between 11 and 13 years?
d) Calculate the 80th percentile.
e) Calculate the 7th percentile.
PROBLEM 4:
Scores of high school seniors taking the English Regents examination in New York State are
normally distributed with a mean of 70 and a standard deviation of 10. Find the probability that a
randomly selected high school senior will have a score between 70 and 75?
PROBLEM 5:
Science scores for high school seniors in the United States are normally distributed with a mean of 60
and a standard deviation of 15. Students scoring in the top 3% are eligible for a special prize
consisting of a laptop and $5,000. What is the approximate cutoff score a student must get in order to
receive the prize?
PROBLEM 6:
The average grade point average (GPA) of undergraduate students in New York is normally
distributed with a population mean of 2.5 and a population standard deviation of .5. Compute
the following, showing all work:
(I) The percentage of students with GPA's between 1.3 and 1.8 is: (a) less than 5.6%
(b) 5.7% (c) 5.9%
(d) 6.2% (e) 6.3%
(f) 6.6% (g) 7.3% (h) 7.5%
i) 7.9% (j) more than 8%.
(II) The percentage of students with GPA's above 3.6 is: (a) less than 1% (b) 1.2% (c)
1.4% (d) 1.6%
(e) 1.9% (f) 2.2% (g) 2.5% (h) 2.7% (i) 3.0% (j) more than 3%.
(III) Above what GPA will the top 3% of the students be (i.e., compute the 97th percentile):
(a) less than 2.98
(b) 2.98
(c) 3.04
(d) 3.14
(e) 3.22 (f) 3.31 (g) 3.44
(h) 3.57
(i) 3.64
(j) more than 3.64.
(IV) If a sample of 25 students is taken, what is the probability that the sample mean GPA will
be between 2.50 and 2.75? (a) less than .10 (b) .122
(c) .243 (d) .307
(e) .346
(f) .38 (g) .42 (h) .44 (i) .494
the central limit theorem.
(j) more than .494. To do this problem, you must know
PROBLEM 7: Life of a GE Stove
The average life of a GE stove is 15.0 years (population mean) with a population standard
deviation of 2.5 years.
(a) What percentage of GE stoves will last 10 years or less?
(b) What percentage of GE stoves will last 18 years or more?
(c) What percentage of GE stoves will last between 16 and 20 years?
(d) Calculate the 1st percentile
(e) Calculate the 96th percentile
PROBLEM 8: The average wage of plumbers
The average hourly wage of plumbers is normally distributed with a population mean of
$24.00 and a population standard deviation of $6.00. Calculate the following:
(a)
(b)
(c)
(d)
The proportion of plumbers earning between $18 and $22
The proportion of plumbers earning more than $28
The proportion of plumbers earning less than $15
The 70th percentile
These problems require that you read and understand the central limit theorem.
PROBLEM 9.
A laptop manufacturer finds that the average time it takes an employee to load a laptop with
software is 50 minutes with a standard deviation 20 minutes. Suppose you take a random
sample of 100 employees. The standard deviation of the sample mean is:
PROBLEM 10.
A company that manufactures bookcases finds that the average time it takes an employee to
build a bookcase is 25 hours with a standard deviation of 9 hours. A random sample of 81
employees is taken. What is the likelihood that the sample mean will be 26 hours or more?