1. Site preparation for a new housing development with tracts for several homes is
normally distributed with a mean of 200 workdays and a standard deviation of 10
workdays.
a. Find the probability that the site is completed within 215 workdays. _________
b. Find the number of workdays that represents a probability of 0.99 that the site is
completed. _______
c. Find the probability that the site is completed between 205 and 215 workdays.
________
2. The table below presents the probability distribution for X = the number of email
messages you will receive in the next 10 minutes:
X P(X)
0 .35
1 .35
2 .25
3 .05
a. What is the probability that no emails will be received? _____
b. What is the probability that at least one email will be received? ______
c. What is E(X), the expected number of emails that will be received? _____
d. What is V(X), the variance in the number of emails that will be received? _____
3. The Census Bureau has found that 26% of U.S. families are headed by a single
parent. If 80 families are chosen at random to be interviewed:
a. What is the expected number of families headed by a single parent? _______
b. What is the standard deviation of the number of families that are headed by a single
parent? _____
c. What is the expected proportion of families headed by a single parent? ______
d. What is the standard deviation of the proportion of families headed by a single
parent? ____
e. Find the approximate probability that at least 25 of the families are headed by a single
parent, using the normal distribution and the continuity correction. _______
f. Suppose that 33 families in your sample are each headed by a single parent; is this
result surprising? Why?
4. Suppose that you are the owner of a service station in Seattle where gasoline
customers arrive randomly on weekday afternoons at an average of 3.2 every 4
minutes.
a. What is the expected number of arrivals in a 4-minute interval on a weekday
afternoon? ______
b. What is the standard deviation of the number of arrivals in a 4-minute interval on a
weekday afternoon? ______
c. What is the probability of having more than 1 customer in a 4-minute interval on a
weekday afternoon? _______
BA 500- Statistics-Sample Exam 1- Pilcher- page 1 of 5
5. The Environmental Protection Agency is trying to decide which of two industrial sites
to investigate. The regional director estimates that the probability of a federal law
violation is 0.3 for the first site and 0.25 for the second site. The director also believes
that the occurrences of violations at the two sites are mutually exclusive.
a. Find the probability of federal law violation at the first site or the second site or both
sites. ______
b. Given there is a federal law violation at the first site, find the probability that there is
also a federal law violation at the second site. _______
6. The Manager of Operations at an assembly plant is analyzing employee absenteeism.
Ten percent of all plant employees work in the finishing department, 20% of all plant
employees are absent excessively, and 7% of all plant employees work in the
finishing department and are absent excessively.
a. Find the probability that an employee selected at random works in the finishing
department. ____
b. Find the probability that an employee selected at random works in the finishing
department and is absent excessively. ________
c. Find the probability that an employee selected at random either works in the finishing
department or is absent excessively. ________
d. Given that the employee selected at random works in the finishing department, find
the probability that the employee is absent excessively. _____
e. Given that the employee selected at random is absent excessively, find the probability
that the employee works in the finishing department. ____
f. Are the events “working in the finishing department” and “absent excessively”
mutually exclusive? ___
g. Are the events “working in the finishing department” and “absent excessively”
independent? ___
h. If the percentage of employees who worked in the finishing department and were
absent excessively was found to be 2% instead of the original 7%, are the events
“works in the finishing department” and “absent excessively” independent?
BA 500- Statistics-Sample Exam 1- Pilcher- page 2 of 5
7. A mail-order company has estimated sales for a popular item to be normally
distributed with a mean of 180,000 units and a standard deviation of 15,000 units.
a. What is the probability of selling all the units on hand if the inventory equals 200,000
units? _________
b. What inventory should the company have on hand if they want the probability of
running out of stock to be 5%? _________
c. What is the optimal stocking level if overstocking costs $2 per item and under-stocking
costs $4 an item? _________
8. The manufacturer of a commercial television monitor guarantees the picture tube for
one year (8760 hours = 24 hours a day * 365 days a year). The monitors are used in
airport terminals for flight schedules, and they are continuously in use with the power
on. The mean life for these tubes is 20,000 hours, and their time to failure has an
exponential distribution. It costs the manufacturer $200 to manufacture, sell, and
deliver a monitor that is sold for $300. It costs $150 to replace a failed tube, including
materials and labor.
a. What is the probability that a monitor must be replaced under warranty? _________
b. How likely is it that a monitor lasts less than or equal to the mean life of 20,000 hours?
_________
c. What is the manufacturer’s expected profit? _________
BA 500- Statistics-Sample Exam 1- Pilcher- page 3 of 5
Formula List- BA 500- Statistics- Exam 1- Pilcher
the average of a sample of size n
1 n
X  X 2 ... X n
X   Xi  1
n i 1
n
the average of a population of size N
1

N
N
X
i 1
i

X 1  X 2 ... X N
N
the weighted average of a sample of size n
n
 w1 X 1  w2 X 2 ... wn X n   wi X i
i 1
n
s
(X
i 1
 X )2
for sample data
n1
N

i

i 1
( X i  )2
for population data
N
sample coefficient of variation:
s
X
population coefficient of variation:


two events A, B are independent if Prob(A) = Prob (A\B)
or, equivalently,
Prob(B) = Prob (B\A)
or, equivalently,
Prob(A and B) = Prob(A)*Prob(B)
Prob ( A \ B) = Prob ( A and B) / Prob (B)
original add d
average X
standard S
deviation
multiply by c add and multiply
cX
cX  d
X d
S
cS
BA 500- Statistics-Sample Exam 1- Pilcher- page 4 of 5
cS
z
( X  )

Mean or Expected Value of Random Variable X:
  E ( X )   X * P( X )
Standard deviation of Random Variable X:

 ( X  )
2
P( X )
The probability function for the Binomial:
 n
P( X  a )     a (1  ) n  a
 a
 n
n!
 a  a !(n  a )!
The probability function for the Poisson is:
a e  
for a =0,1,2,3,4,5,6....., where e = 2.71828 (approx.)
P( X  a ) 
a!
For the exponential distribution with a mean of mu (  ), the cumulative distribution
function is:
where   
P(t  a)  1  e
a
for a>0.
Random Variable
___________________________________________________
Number of Occurrences, X
Proportion or Percentage, p=X/n
__________________________________________________________________________
Standard Deviation
(for the population)
 X  n (1   )
p 
 (1   )
n
Standard error
(estimated from a sample) )
S X  np(1  p)
Sp 
p(1  p)
n
_________________________________________________________________________
Random Variable
__________________________________________________
Average
Sum Total
_________________________________________________________________________
 sum  n
x  
Mean
Standard deviation
Standard error
X 
SX 

n
 sum   n
S
n
_________________________________________________________________________
BA 500- Statistics-Sample Exam 1- Pilcher- page 5 of 5