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STA 2023 - Quiz 2 (Answer Key)
Spring 2003
Quiz proceedings:
 No cheating
 Show all relevant work to receive full credit – numbers in parentheses are point values for each problem
 Please turn off your cellular telephone
 Not permitted for use during the quiz: HyperColor T-shirts, Pop Rocks® with soda, Britney Spears CDs
 Good luck!
1. A discrete probability distribution is defined as follows: 4.7, 4.23
X
P(X)
-1
.10
0
1
.54
2
.28
a. Calculate P(0). (3)
p(0)  1  (.10  .54  .28)  1  .92  .08
b. Calculate E(X). (3)
E( x)     x  p( x)  (1)(.10)  (0)(.08)  (1)(.54)  (2)(.28)  .10  0  .54  .56  1
c. Calculate 2. (3)
 2   ( x   ) 2 p( x)  (1  1) 2 (.10)  (0  1) 2 (.08)  (1  1) 2 (.54)  (2  1) 2 (.28)
 (2) 2 (.10)  (1) 2 (.08)  (0) 2 (.54)  (1) 2 (.28)  .76
d. Calculate P(X>0). (3)
P( x  0)  P( x  1)  P( x  2)  .54  .28  .82
e. What is the probability that x is less than 1.5? (3)
P( x  1.5)  P( x  1)  P( x  0)  P( x  1)  .10  .08  .54  .72
2. Records indicate that the daily high temperature on the island of Nauru tend to have a uniform
distribution over the interval from 70 F to 95 F. A tourist arrives on the island on a randomly selected
day in January. 5.7
a. What is the expected temperature? (2)
c  d 70  95 165



 82.5
2
2
2
b. What is the probability that the temperature will be under 80 F? (3)
1
1
1
f ( x) 


d  c 95  70 25
 1 
 1 
P(x  80)   80  70    10  .4
 25 
 25 
c. What is the probability that the temperature will be between 85 F and 90 F? (3)
 1 
 1 
P(85  x  90)   90  85   5  .2
 25 
 25 
3. The number of corny phrases (i.e. “…shines like a beacon of freedom”) made up by George W. Bush
during any particular speech is distributed as a Poisson random variable with an average of 1.5. 4.64
a. Calculate the value of . (2)
    1.5
b. What is the probability that President Bush does not use any corny phrases during a particular
speech? (5)
P( x  0)  P( x  0)  .223 (from Table III)
c. What is the probability that President Bush uses more than two corny phrases during a particular
speech? (5)
P( x  2)  1  P( x  2)  1  .809  .191 (.809 from Table III)
4. The assumed travel times of successive taxi trips to and from an airport are distributed as independent
exponential random variables. Assume  = 30 minutes. 5.112
a. What is the mean trip time for the taxi service? (3)
    30
b. What is the probability that a particular trip will take more than 30 minutes? (3)
P( x  30)  e

30
30
 e 1  .367879 (from Table V)
c. What is the probability that a particular trip will take less than 15 minutes? (3)
P( x  15)  1  e

15
30
 1  e .5  1  .606531  .393469 (.606531 from Table V)
d. Two taxis have just been dispatched. What is the probability that both will be gone for more
than 30 minutes? (3)
P( x1  30  x2  30)  P( x1  30)  P( x2  30)  (.367879)(.367879)  .135335
5. Suppose you are over 21 years of age and you are purchasing cases of wine (twelve bottles per case) and
that, periodically, you select a test case to determine the adequacy of the bottles’ seals. To do this, you
randomly select and test three bottles in the case. Assume a particular case that you inspect contains one
spoiled bottle of wine. 4.85
a. What is the average number of spoiled bottles of wine that will turn up in your sample? (4)
nr 3  1 3



 .25
N
12 12
b. What is the probability that the spoiled bottle will turn up in your sample? (4)
111
  
1 2
1  55 55
P(x  1)     

 .25
220 220
12 
 
3
6. Consider an experiment where we have 15 people all tossing a fair quarter. Let x represent the number
of tails that appear. 4.43, 4.51
a. Calculate the mean of x. (3)
  np  (15)(. 5)  7.5
b. What is the probability that you we observe exactly 7 tails? (4)
P( x  7)  P( x  7)  P( x  6)  .500  .304  .196 (.500 and .304 from Table II)
c. What is the probability that you observe no more than 4 tails? (4)
P(x  4)  .059 (from Table II)
d. Calculate P(4 < x < 10). (4)
P(4  x  10)  P( x  9)  P( x  4)  .849  .059  .79 (.849 and .059 from Table II)
7. It is known that the national percentage of college students who vote in their school’s student
government elections is 20%. Suppose that we wish to predict the student voting results at UCF
(vote/not vote), using the national average. The enrollment at UCF is approximately 40,000. 5.59
a. Calculate the mean of x, where x represents the number of students who vote. (2)
  np  (40,000)(.20)  8,000
b. Calculate the standard deviation of x, where x represents the number of students who vote. (2)
  npq  (40,000)(.20)(.80)  6,400  80
c. Will the normal approximation to the binomial be appropriate in this situation? Explain. (2)
  3  8,000  (3)(80)  8,000  240  (7760,8240)
(7760,8240)  (0,40000) The normal approximation to the binomial is appropriate.
d. Regardless of your answer in part c, find the probability that no more than 8,200 students will
vote in the election. (4)
8200.5  8000
P( x  8200)  P( z 
)  P( z  2.51)  .5  .4940  .9940 (.4940 from Table IV)
80
8. Let z represent the standard normal distribution (as always). 5.17
a. What are the mean and standard deviation of the standard normal distribution? (1)
=0, =1
b. Calculate P(0 < z < 2.05). (3)
P(0  z  2.05)  .4798 (from Table IV)
c. Calculate P(z > 2.5). (3)
P( z  2.5)  .5  P(0  z  2.5)  .5  .4938  .0062 (.4938 from Table IV)
d. Find the value of z that has (approximately) 10% above it. (3)
P( z  z 0 )  .10  P(0  z  z 0 )  .40  z 0  1.28 (from Table IV)
9. Let x represent the grade point averages of the students at a certain university. It is known that x is
normally distributed with  = 2.5 and  = .4. 5.24, 5.25
a. Draw a graph of the distribution of x. If you have time, make it look pretty. (3)
x
1.3
1.7
2.1
2.5
2.9
3.3
3.7
b. What is the probability that a randomly selected student has a GPA above 3.5? (3)
3.5  2.5
P( x  3.5)  P( z 
)  P( z  2.5)  .0062 (from question 8c or Table IV)
.4
c. The procedure for determining individuals graduating with honors is to award those students
with GPAs in the top 10%. What should the corresponding cutoff GPA be? (3)
P( x  x0 )  .10  P(0  x  x0 )  .40  P(0  z  z 0 )  .40  z 0  1.28
x    z  2.5  (1.28)(.4)  2.5  .512  3.012
10. Assume x = you, and x can spend an evening with y, where y is any person, living or dead, real or makebelieve. Assuming equal population variances, what is your choice for y? (1)



Assume that any dead person comes back to life for this one evening (more than just a corpse)
This question is not intended to be sexual, so please omit references to specific sexual anatomy
If you have time, state reasons for your choice of y
~ The End ~