# Stat350Midterm1

```Statistics 350 Midterm 1
Time: 1 hour
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
may be taking it at a different time. I have not been told anything about the exam by someone
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
A) True
B) False
2) The standard deviation of the zi is 1
A) True
B) False
3) The zi ’s histogram is bell-shaped.
A) True
B) False
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)
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)=_________________, E(X-Y)=______________________
Var(X+Y)=________________, Var(X-Y)=_______________________
Var(X+X)=__________________, Var(X-X)=____________________________
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 =_______________
3) Suppose a random variable X follows a distribution of Binomial(4, 0.9),
the median is ______________(&lt;, &gt; 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 ?
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
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 ?
d) (3pt) In this population, what is the bone mineral density such that only 3% of the population
has a lower level?
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 .
b) (2pt) Find the probability that X is greater than 0.5.
c) (2pt) Find the mean of X .
d) (2pt) Find the standard deviation of X .
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.
f)
(4pt) Consider a randomly selected sample of size 20, find the proability that the sample
mean is inclusively within 0.01 of 0.6
5 out of 5
```