Statistics 350 Midterm 1
Time: 1 hour
Name (please print): ___________________________
Show all your work and calculations. Keep 2 decimal places and circle your final answer.
Partial credit will be given for work that is partially correct. Points will be deducted for false
statements shown in the solution to a problem, even if the final answer is correct.
This exam is closed-book. You may not consult any notes or books during this exam. Only a
non-graphical calculator, a formula sheet and relevant tables are allowed.
Please sign below to indicate your agreement with the following honour code.
Honour code: I promise not to cheat on this exam. I will neither give nor receive any
unauthorized assistance. I promise not to share information about this exam with anyone who
may be taking it at a different time. I have not been told anything about the exam by someone
who has already taken it.
Signature:
.
Question
Possible
1
8
2
16
3
10
4
16
Total
50
Actual
Date:
.
1. (20pts) Multiple choice and short calculation problems
Consider a sample x1 , ..., xn with mean x and standard deviation s and let zi 
xi  x
, verify
s
the following statement in 1) , 2) and 3):
1) The mean of zi is 0
Answer: A) True
2) The standard deviation of the zi is 1
Answer: A) True
3) The zi ’s histogram is bell-shaped.
Answer: B) False, z is normal if X is normal
Case study
4) If a random variable X denotes the number of trees in 2-acre plot within a certain
forest, if an average density is 80 trees per acre, then which of the following statement
is true
A)
X ~ Binomial (80, 0.5)
B)
X ~ Binomial (160, 0.5)
C)
X ~ Poisson(80)
D) X ~ Poisson(160)
Answer: D)
1 out of 5
2. Fill in the blanks, 2 points each, no partial credit.
1) If two random variables, X ~ Normal (  ,  ) , Y ~ Normal (  ,  ) ， X and Y are
independent, find the answers to the following questions.
E(X+Y)=_____ 2  ____________, E(X-Y)=_______0_______________
Var(X+Y)=___ 2 2 _____________, Var(X-Y)=_____ _ 2 2 __________
Var(X+X)=___ 4 2 ______________, Var(X-X)=______0_____________
2) Consider a system of components connected as shown in the following figure.
Suppose all components work independently of one another and
Pr(a given component works)=0.9,
the probability that the entire system works correctly =____0.8019_________
Pr(system works)=Pr(A and (BorCorBoth) and D)
3) Suppose a random variable X follows a distribution of Binomial(4, 0.9),
the median is ______>________(<, > or =) the mean of the distribution .
2 out of 5
3.(10pt) The distribution of bone mineral density in women between the ages of 45 and 64 is
normal with mean 0.9 g/ cm 2 and a standard deviation of 0.15 g/ cm 2 .
a). (2pt) What percent of the population has bone mineral density more than 1.2 g/ cm 2 ?
Pr ( X  1.2)  Pr (
X  0.9 1.2  0.9

)  Pr ( Z  2)  0.0228
0.15
0.15
b) (2pt) What is the probability that the bone mineral density of a randomly chosen women is
between 0.9 and 1.2 g / cm2
Pr (0.9  X  1.2)  Pr (
0.9  0.9
1.2  0.9
Z
)
0.15
0.15
= Pr ( Z  2)  Pr ( Z  0)  0.4772
c) (3pt) Suppose that the bone mineral density of a randomly chosen woman is more than 0.9
g / cm2 , what is the probability that her bone mineral density is less than 1.2 g / cm2 ?
Pr (0.9  X  1.2) 0.4772

 0.9544
Pr ( X  0.9)
0.5
d) (3pt) In this population, what is the bone mineral density such that only 3% of the population
has a lower level?
Pr ( X  X 0.03 )  0.03
X 0.03  Z0.03    1.88(0.15)  0.9  0.618
3 out of 5
4. (16pt) A random variable X has a probability density function as follows:
kx 2 (1  x) 0  x  1
f ( x)  {
0
otherwise
a) (2pt) Find the value of k .

1
0
f ( x)dx  1 , solve for k , k =12
b) (2pt) Find the probability that X is greater than 0.5.

1
0.5
f ( x)dx  0.6875
c) (2pt) Find the mean of X .
1
E ( X )   x f ( x)dx  0.6
0
d) (2pt) Find the standard deviation of X .
1
1
1
0
0
0
Var ( X )   ( x   )2 f ( x)dx   x 2 f ( x)dx  ( x f ( x)dx)2  0.04
  .04  0.2
4 out of 5
e) (4pt) Consider a randomly selected sample of size 20, find the probability that at least
one observation is greater than 0.5.
Define Y be the numbe of observations that is greater than 0.5, Y~Binomial(20, 0.6875)
 20 
Pr (Y  1)  1  Pr (Y  0)  1    (0.6875)0 (1  0.6875) 20  1
0
f)
X
(4pt) Consider a randomly selected sample of size 20, find the proability that the sample
mean is inclusively within 0.01 of 0.6
Normal (  ,

20
)  Normal (0.6, 0.0447)
Pr (0.59  X  0.61  Pr (0.22 
 0.22))  0.5871  0.4129  0.1742
5 out of 5