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STAT 211
SUMMER 2002
You have 75 minutes to complete this exam. You can use your own calculator only. There is no partial
credit on this exam. If you did not mark your final answer on your scantron, your answer will be counted
incorrect. If you caught cheating, you will get a grade of zero. Good Luck.
EXAM 1 - FORM A
The weights of samples of six tubers of four varieties of potatoes (BUR, KEN, NOR, RLS) grown under
specific laboratory conditions are as follows.
BUR
KEN
NOR
RLS
0.19
0.35
0.27
0.08
0.00
0.36
0.33
0.29
0.17
0.33
0.35
0.70
0.10
0.55
0.27
0.25
0.21
0.38
0.40
0.19
0.25
0.38
0.36
0.19
The descriptive Statistics and the graphs for this data set is as follows.
Variable
BUR
KEN
NOR
RLS
N
6
6
6
6
Mean
0.1533
0.3917
0.3300
0.2833
Median
0.1800
0.3700
0.3400
0.2200
TrMean
0.1533
0.3917
0.3300
0.2833
Variable
BUR
KEN
NOR
RLS
Minimum
0.0000
0.3300
0.2700
0.0800
Maximum
0.2500
0.5500
0.4000
0.7000
Q1
0.0750
0.3450
0.2700
0.1625
Q3
0.2200
0.4225
0.3700
0.3925
StDev
0.0900
0.0799
0.0518
0.2161
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
BUR
KEN
Answer questions 1 to 11 using the information above.
NOR
RLS
SE Mean
0.0368
0.0326
0.0211
0.0882
1.
Which variety has outliers in the data set?
(a) none of the varieties
(b) BUR
(c) KEN
(d) NOR
(e) RLS
2.
Which of the following is the interquartile range for BUR?
(a) 0.0775
(b) 0.10
(c) 0.145
(d) 0.23
(e) 0.25
3.
Which of the following is correct?
(a) All varieties have the same number of data collected
(b) BUR has more data points than the others
(c) KEN has more data points than the others
(d) NOR has more data points than the others
(e) All varieties have the different number of data collected
4.
Which variety has the most variation in the data?
(a) none of the varieties
(b) BUR
(c) KEN
(d) NOR
(e) RLS
5.
Which variety has the range 0.22?
(a) none of the varieties
(b) BUR
(c) KEN
(d) NOR
(e) RLS
6.
Which variety is more skewed than the others in absolute value?
(a) none of the varieties
(b) BUR
(c) KEN
(d) NOR
(e) RLS
7.
Which of the following is correct?
(a) All varieties have the same skewness
(b) Exactly one variety is negatively skewed whereas the others are positively skewed
(c) Exactly two varieties are negatively skewed whereas the other two are positively skewed
(d) Exactly three varieties are negatively skewed whereas the other is positively skewed
(e) None of the varieties have the same skewness
8.
What percent of the data for KEN is larger than its mean value?
(a) 0%
(b) 16.67%
(c) 50%
(d) 83.33%
(e) 100%
9.
Consider all data points. What percent of the weights are at most 0.19?
(a) 11.67%
(b) 16.67%
(c) 29.17%
(d) 23.17%
(e) 31.69%
10. Which variety has the median 0.34?
(a) none of the varieties
(b) BUR
(c) KEN
(d) NOR
(e) RLS
11. Which of the following is the coefficient of variation for KEN?
(a) 15.697
(b) 20.398
(c) 50.000
(d) 58.708
(e) 76.280
12. Which of the following cannot be qualitative data?
(a) name of the medicine grandmother uses.
(b) name of the cars observed on Rudder freeway everyday
(c) name of the cities
(d) number of cars observed on Rudder freeway everyday
(e) all of the above
13. In a sample of n=100 observations, the density of a histogram class interval of width 4 is 0.01. Which
of the following is the frequency for the same interval?
(a) 2
(b) 4
(c) 10
(d) 20
(e) 40
14. If each component works with probability 0.9, which of the following is the probability for the system
to work?
_________ ______2 _________________
/
\
_________ 1 ________/
\______5_____ 6 ________
\
/
\___________3______ ___ 4 __________ /
(a)
(b)
(c)
(d)
(e)
0.5314
0.6173
0.7152
0.7217
0.8971
Among the 1820 subjects in a study, 30 suffered from tuberculosis and 1790 did not. Chest x-rays were
administered to all individuals; 73 had a positive x-ray-implying that there was significant evidence of
inflammatory disease-whereas the results the other 1747 were negative. The data for the study are
presented in the table below. Answer questions 15 to 19 using this information.
No Tuberculosis
Tuberculosis
Total
Negative x-ray
1739
8
1747
Positive x-ray
51
22
73
Total
1790
30
1820
15. What is the probability that a randomly selected individual has tuberculosis?
(a) 0.0044
(b) 0.0165
(c) 0.0121
(d) 0.2667
(e) 0.7333
16. What is the probability that a randomly selected individual has positive x-ray given that he or she has
tuberculosis?
(a) 0.2667
(b) 0.3014
(c) 0.6986
(d) 0.7333
(e) 0.9555
17. What is the probability that a randomly selected individual has tuberculosis given that his or her x-ray
is positive?
(a) 0.2667
(b) 0.3014
(c) 0.6986
(d) 0.7333
(e) 0.9555
18. Consider the events of an individual suffering from tuberculosis (A) and not suffering from
tuberculosis (B). Are these events mutually exclusive?
(a) Yes, P(AB)=0
(b) Yes, P(AB)=P(A)P(B)
(c) No, P(AB)0
(d) No, P(AB)P(A)P(B)
(e) No, P(A|B)=P(A)
19. Consider the events of an individual suffering from tuberculosis (A) and not suffering from
tuberculosis (B). Are these events independent?
(a) Yes, P(AB)=0
(b) Yes, P(AB)=P(A)P(B)
(c) No, P(AB)0
(d) No, P(AB)P(A)P(B)
(e) No, P(A|B)=P(A)
20. A certain system can experience three different types of defects. Let Ai (i=1,2,3) denote the event that
the system has a defect of type i. Which of the following is the system having only type 2 and type 3
defects but not type 1.
(a) A1 A2A3
(b) A1 A'2A3
(c) A'1 A2A3
(d) A1 A2A'3
(e) A1 A'2A'3
Let X be a discrete random variable that represents the number of diagnostic services a child receives
during an office visit to a pediatric specialist; these services include procedures such as blood tests and
urine analysis. The probability distribution for X appears below.
x
0
1
2
3
4
5
Total
P(X=x)
0.671
0.229
0.053
0.031
0.01
0.006
1
Use the information above to answer questions 21 to 23.
21. What is the probability that a child receives exactly three diagnostic services during an office visit to a
pediatric specialist?
(a) 0.031
(b) 0.037
(c) 0.044
(d) 0.956
(e) 0.963
22. What is the probability that he or she receives at least one service?
(a) 0.1
(b) 0.329
(c) 0.671
(d) 0.9
(e) 1
23. What is the expected number of diagnostic services a child receives during an office visit to a
specialist?
(a) 0.1667
(b) 0.498
(c) 0.502
(d) 0.8333
(e) 2.5
Let X be a random variable that represents the number of infants in a group of 2000 who die before
reaching their first birthdays. In the United States, the probability that a child dies during his or her first
year of life is 0.0085. Use this information to answer questions 24 and 25.
24. What is the expected number of infants who would die in a group of this size?
(a) 7
(b) 9
(c) 17
(d) 20
(e) 23
25. What is the probability that at least one infant out of 2000 die in their first year of life?
(a) 0
(b) 0.0085
(c) 0.9915
(d) 1
(e) need more information to compute it
Answer Key:
1.c
2.c
13.b
14.c
25.d
3.a
15.b
4.e
16.d
5.c
17.b
6.b
18.a
7.c or e 8.b
19.d
20.c
9.c
21.a
10.d
22.b
11.b
23.b
12.d
24.c
Some of the formulas that you may need
Relative Frequency = frequency / sample size.
The Range: R = the largest data - the smallest data
Density = Relative Frequency / class width
The Interquartile range: IQR = Q3 - Q1
_
3  ( x  median)
Coefficient of Skewness: SK=
s
_
Coefficient of variation: CV=100( s / x )
_
The sample mean of X's:
x   xi / n . The sample variance of X's: s 2 
x
2
i
i
n 1
i 1
The sample standard deviation of X's:
_
 n  ( x) 2
s  s2
Mutually exclusive or Disjoint: Two events, A and B have no outcomes in common.
Conditional Probability: For any two events A and B with P(B)>0, the conditional probability of A given
B has occurred is defined by P(A | B ) = P(AB) / P(B).
Independence: Two events A and B are independent if P(AB) = P(A)P(B)
Predicting the Reliability of Systems with n components:
Ri(t): The reliability of individual component = P(component i survives beyond time t)
Rs(t): The reliability of the system = P(system works)
Series System

Rs(t)=R1(t)R2(t)…….Rn(t)
Parallel System 
Rs(t) = 1 - [1-R1(t)][1-R2(t)]……[1-Rn(t)]
Expected value for the discrete random variable, X: Weighted average of the possible values. Expected
value of the random variable X,
  E( X ) 
 x  p ( x)
for all x
Variance

=
x
for
2
 for all x
the
discrete
random
X: 
variable,
2
 Var ( X )  E ( X 2 )  [ E ( X )] 2

 p( x)   [ E ( X )] 2

Discrete probability distribution, p(x) is legitimate if
 0  p(x)=P(X=x)  1 for all x where X is a discrete random variable (r.v).

 p ( x)  1
all x
Cumulative Distribution Function, (CDF) for discrete X:
F ( x)  P( X  x) 
x
 P( X  y )
for all y
P(a  X  b)  F (b)  F (a  1) and P( X  a)  F (a)  F (a  1) where a and b are
The expected value of a discrete variable, X :
 x  p ( x)
  E( X ) 
for all x
The variance of a discrete variable, X: 2 = Var(X) =
E( X ) 
2
x
2
 p ( x)
for all x
The standard deviation of a discrete variable, X:  =
2
E[( X   ) 2 ]  E ( X 2 )   2
where
Bernoulli Distribution: It is based on Bernoulli trial (an experiment with two, and only two, possible
outcomes). A r.v. X has a Bernoulli(p) distribution if
P(X=x)=
p x  (1  p)1x ,
x  0,1 and 0p1. where E(X) = p and Var(X) = p(1-p)
Binomial Distribution: Approximate probability model for sampling without replacement from a finite
dichotomous population. X~Binomial(n,p).
 n fixed trials
 each trial is identical and results in success or failure
 independent trials
 the probability of success (p) is constant from trial to trial
 X is the number of successes among n trials
n
P( X  x)     p x  (1  p) nx ,
 x
E(X) = np
and
Var(X) = np(1-p)
x  0,1,2,...., n