1. For each question below indicate True (T) or False (F)
a. The binomial distribution is a possible model for a continuous variable: F
b. In any normal distribution 95% of the probability lies within two standard deviations of the mean: T
c. For a Poisson(m=4) distribution the variance is 2: F
d. For any exponential distribution, the mean is greater than the median: T
e. The Poisson is a good approximation to binomial when n is large and p is small. T
(2+2+2+2+2=10 points)
2. Given that the area under the standard normal curve, to the left of –2.3 is .0107, what is the area
under the normal curve to the right of 2.3?
(show work) DTDP ____0.0107____________
value
(8 points)
3. Suppose you flip a fair coin 7 times, let X be the possible number of heads. Find the following
probabilities (in each case show work below):
(i) P(X = 0) =___(.5)7______________ (ii) P(X = 1) = __7*.5*.56_________
(value) (value)
(iii) Probability of at least 2 heads: Prob. Statement: _P(X > 2)__ value __1-(.5)7-7*(.5)7___
(5+5+7+5=22 points)
4. You are the safety inspector at some parts manufacturing plant. Safety at the plant is a concern; it is
known that on an average there are 5 accidents per week. Assuming that the number of accidents in
any week follows a Poisson distribution with mean 5, what's the probability that in 2 weeks there will
be only one accident? Let X be the number of accidents in 2 weeks.
______P(X=1)________________ __10*e-10__________
Prob. Statement value
(show work: Hint: what's the distribution of X?)
X~Poisson(mean=2*5=10) (8+7=15 points)
5. The scores on a test are normally distributed with a mean of 80 and a standard deviation of 5. The
score distribution is shown in figure 1 below. Answer the following questions. Let X denote the
variable score.
(a) Refer to the blue shaded area in figure 1. This is the probability of:
__P(X < 70)______________ (just write the probability statement).
(b) Find the value of probability in part (a) (show work)
_P(Z<-2) = .0228__________
(c) What's the probability of scoring between 70 and 80?
(i) Draw the picture: Label and Shade the area to indicate the required area.
P(70<X<80) =P(-2<Z<0)=.5-.0228=.4772 --- DTDP
(ii) The value of the required probability in (i) is _______.4772______________.
(show work below) value
(d) What's the probability of scoring 95 or above?
_______.0013________________
(show work below) value
P(X > 95) =P(Z>3)=P(Z<-3)=.0013
(5+7+7+6=25 points)
6. The waiting time at a local restaurant follows an exponential distribution with a mean of 5 minutes.
Answer the following questions (in each case show work below)
(i) What's the probability of waiting for more than 5 minutes at this restaurant?
_____P(X > 5)__________________ _______e-5/5=e-1=.3679_____________
Prob. Statement value
(ii) What's the probability of waiting for more than 2 but less than 4 minutes?
_____P(2 < X < 4)_________________ _e-2/5-e-4/5=.6703-.4493=.2210_______
Figure 1
70
Prob. Statement value
(5+5+5+5=20 points)