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March 5, 2012
AP Statistics
Unit 5 Test (Ch.7 & 8)
Multiple Choice
1. A basketball player makes 160 out of 200 free throws. We would estimate the
probability that the player makes his next free throw to be
a.
b.
c.
d.
e.
0.16
0.50
0.80
1.2
None of these
2. A dealer in the Sands Casino in Las Vegas selects 40 cards from a standard deck of 52
cards. Let Y be the number of red cards (hearts or diamonds) in the 40 cards selected.
Which if the following best describes this setting:
a. Y has a binomial distribution with n=40 observations and probability of success
p=0.5
b. Y has a binomial distribution with n=40 observations and probability of success
p=0.5, provided the deck is shuffled well.
c. Y has a binomial distribution with n=40 observations and probability of success
p=0.5, provided after selecting a card it is replaced in the deck and the deck is
shuffled well before the next card is selected.
d. Y has a normal distribution with mean p = 0.5
e. None of these
3. A random variable is
a.
b.
c.
d.
e.
a hypothetical list of the possible outcomes of a random phenomenon.
any phenomenon in which outcomes are equally likely.
any number that changes in a predictable way in the long run.
a variable whose value is a numerical outcome of a random phenomenon.
None of these
4. In a certain population, 40% of households have a total annual income of over
$70,000. A simple random sample is taken of 4 of these households. Let X be the number
of households on the sample with an annual income of over $70,000 and assume that the
binomial assumptions are reasonable. What is the mean of X?
a.
b.
c.
d.
e.
1.6
28,000
0.96
2, since the mean must be an integer
The answer cannot be computed from the information given.
5. The probability that a three-year-old battery still works is 0.8. A cassette recorder
requires four working batteries to operate. The state of the batteries can be regarded as
independent, and four three-year-old batteries are selected for the cassette recorder. What
is the probability that the cassette recorder operates?
a.
b.
c.
d.
e.
0.9984
0.8000
0.5904
0.4096
The answer cannot be computed from the information given.
6. Which of the following random variables should be considered continuous?
a.
b.
c.
d.
The time it takes for a randomly chosen woman to run 100 meters.
The number of brothers a randomly chosen person has.
The number of cars owned by a randomly chosen adult male.
The number of orders received by a mail order company in a randomly chosen
week.
e. None of these
7. Twenty percent of all trucks undergoing a certain inspection will fail the inspection.
Assume the trucks are independently undergoing this inspection, one at a time. The
expected number of trucks inspected before a truck fails this inspection is
a.
b.
c.
d.
e.
2
4
5
20
The answer cannot be computed from the information given.
8. In a population of students, the number of calculators owned is a random variable X
with P(X = 0) =.2, P(X = 1) = .6, and P(X = 2) = .2. The mean of this probability
distribution is
a.
b.
c.
d.
e.
0
2
1
0.5
None of those
9. The financial aid office at a state university conducts a study to determine the total
student costs per semester. All students are charged $4500 for tuition. The mean cost for
books is $350 with a standard deviation of $65. The mean outlay for room and board is
$2800 with a standard deviation of $380. The mean personal expenditure is $675 with a
standard deviation of $125. Assuming independence among categories, what is the
standard deviation of the total student costs?
a.
b.
c.
d.
e.
$24
$91
$190
$405
$570
10. Two percent of the circuit boards manufactured by a particular company are
defective. If the circuit boards are randomly selected for testing, the probability that the
number of circuit boards inspected before a defective board is found is greater than 10 is
a.
b.
c.
d.
e.
1.024 x 10^7
5.12 x 10 ^7
0.1829
0 .8171
The answer cannot be computed from the information given.
11. In a particular game, a fair die is tossed. If the number of spots showing is either 4 or
5 you win $1, if the number of spots showing is 6 you win $4, and if the number of spots
showing is1, 2, or 3 you win nothing. Let X be the amount that you win. The expected
value of X is
a.
b.
c.
d.
e.
$0
$1.00
$2.50
$4.00
None of these
12. In a large population of college students, 20% of the students have experienced
feelings of math anxiety. If you take a random sample of 10 students from this
population, the probability that exactly 2 students have experienced math anxiety is
a.
b.
c.
d.
e.
0.3020
0.2634
0.2013
0.5
0.1
13. Refer the previous problem. The standard deviation of the number of students in the
sample who have experienced math anxiety is
a.
b.
c.
d.
e.
0.0160
1.265
0.2530
1
0.2070
14. Let the random variable X represent the profit made on a randomly selected day by a
certain store. Assume that X is normal with the mean $360 and a standard deviation $50.
What is the value of P(X > $400)?
a.
b.
c.
d.
e.
0.8459
0.7881
0.2881
0.2119
The answer cannot be computed.
15. Which of the following are true statements?
I.
II.
III.
a.
b.
c.
d.
e.
The expected value of a geometric random variable is determined by the
formula (1  p)n 1 p .
If X is a geometric random variable and the probability of success is .85, then
the probability distribution of X will be skewed left, since .85 is closer to 1
than to 0.
An important difference between binomial and geometric random variables is
that there is a fixed number of trails in a binomial setting, and the number of
trials varies in a geometric setting.
I only
II only
III only
I, II, and III
None of the above gives the complete set of true responses.
16. A small store keeps track of the number X of customers that make a purchase during
the first hour that the store is open each day. Based on the records, X has the following
probability distribution.
X
0
p(X) 0.1
1
0.1
2
0.1
3
0.1
4
0.6
The standard deviation of the number of customers that make a purchase during the first
hour that the store is open is
a.
b.
c.
d.
e.
1.4
2.0
3.0
4.0
None of these.
17. Suppose there are 3 balls in a box. On one of the balls is the number 1, on another is
the number 2, and on the third is the number 3. You select two balls at random and
without replacement from the box and note the two numbers observed. The sample space
S consists of the three equally likely outcomes {(1,2), (1,3), (2,3)}. Let X be the total of
the two balls selected. Which of the following is the correct set of probabilities for X?
(a) _X_____1_____2_____3__
P(X)
1/3
1/3
1/3
(b) _X_____3_____4_____5__
P(X)
1/3
1/3
1/3
(c) _X_____1_____2_____3__
(PX) 1/6
2/6
3/6
(d) _X_____3_____4_____5__
P(X)
1/6
2/6
3/6
18. A factory makes silicon chips for use in computers. It is known that about 90% of the
chips meet specifications. Every hour a sample of 18 chips is selected at random for
testing. Assume a binomial distribution is valid. Suppose we collect a large number of
these samples of 18 chips and determine the number meeting specifications in each
sample. What is the approximate mean of the number of chips meeting specifications?
a.
b.
c.
d.
e.
16.20
1.62
4.02
16.00
The answer cannot be computed from the information given.
19. The weight of medium-sized tomatoes selected at random from a bin at the local
supermarket is a random variable with mean µ= 10 ounces and standard deviation σ = 1
ounce. Suppose we pick two tomatoes at random from the bin. The difference in the
weights of the two tomatoes selected is a random variable with a mean and standard
deviation (in ounces) of
a.
b.
c.
d.
e.
f.
0, 1
20, 1
0, 1.41
20, 1.41
0, 2
20, 2
20. In a certain population, 40% of households have a total annual income of over
$70,000. A simple random sample is taken of 4 of these households. What is the
probability that 2 or more of the households in the survey have an annual income of over
$70,000?
a.
b.
c.
d.
e.
0.3456
0.4000
0.5000
0.5248
The answer cannot be computed from the information given.
Free Response
Please show all work and computations!!!!
21. The CFO of a trucking firm believes their fleet of trucks, on cross-country hauls, has
a mean of 12.4 miles per gallon with a standard deviation of 1.2 miles per gallon. If this
is a normal distribution
a) What is the probability that one of the trucks averages fewer than 10 mpg?
Answer:
z
10  12.4
 2.0
1.2
P( X  10)  0.0228
b) What is the probability that one of the trucks averages between11.6 and14 mpg?
11.6  12.4
 0.6667
1.2
14  12.4
z
 1.3333
1.2
z
Answer:
P(11.6  X  14)  0.6563
22. Describe the similarities and differences between a binomial and geometric
distribution.
Similarities
Independent Observations
Constant probability of success
Only 2 possible outcomes
Difference
Binomial: fixed # of observations
Geometric: # of observations varies
23. ACT scores for the 1,171,460 members of the 2004 high school graduating class who
took the test closely followed the normal distribution with mean 20.9 and standard
deviation 4.8. Choose two students independently and at random from this group.
a) What is the expected sum of their scores?
Expected (Mean) Sum:  X  X   X   X  20.9  20.9  41.8
b) What is the expected difference of their scores?
Expected (Mean) Difference:  X  X   X   X  20.9  20.9  0
c) What is the standard deviation of the difference in their scores?
Standard Deviation (Difference):
 2 X  X   2 X   2 X  4.8 2  4.8 2  46.08
 X  46.08  6.7882
d) Find the probability that the sum of their scores is greater than 50. Show your method.
Answer:
z
50  41.8
 1.208
6.7882
P( sum  50)  0.1135
24. A headache remedy is said to be 80% effective in curing headaches caused by simple
nervous tension. An investigator tests the remedy on 100 randomly selected patients
suffering from nervous tension.
(a) Define the random variable being measured. X = #of adults who experience
headache relief
(b) What kind of distribution is this? Justify your answer.
Binomial: 2 outcomes – relief, no relief
p is constant = 0.80
fixed # ob observations = 100
independent observations
(c) Calculate the mean and the standard deviation of X.
  np(1  p)
  np
Answer:
  100(0.8)
  80
  100(0.80)(0.20)
 4
(d) Determine the probability that at least 87 subjects experience headache relief
with this remedy.
Answer:
100 
100 
100 
(0.80) 87 (0.20)13  
(0.80) 88 (0.20)12  ...  
(0.80)100 (0.20) 0
P( X  87)  
 87 
 88 
100 
P( X  87)  0.0469
(e) What is the probability that the number of subjects who will obtain relief is
within 1.5 standard deviations of its mean. Justify your method of solution.
Answer:
100 
100 
100 
(0.80) 74 (0.20) 26  
(0.80) 75 (0.20) 25  ...  
(0.80) 86 (0.20)14
P(74  X  86)  
74
75
86






P(74  X  86)  0.8973
(f) Find the probability that the number of subjects who experience headache
relief with this remedy is between74 and 86 using the normal approximation
method.
P(74  X  86)  0.8663
np  10
n(1  p)  10
74  80
Answer: 100(0.80)  10 100(0.20)  10
z
 1.5
4
80  10
20  10
86  80
z
 1.5
4
25. A survey conducted by the Harris polling organization discovered that 63% of all
Americans are overweight. Suppose that a number of randomly selected Americans are
weighed.
(a) Find the probability that 18 or more of the 30 students in a particular adult
class are overweight.
Answer:
 30 
 30 
 30 
P( X  18)   (0.63)18 (0.37)12   (0.63)19 (0.37)11  ...   (0.63) 30 (0.37) 30
 18 
 19 
 30 
P( X  87)  0.7055
(b) How many Americans do you expect to weigh before you encounter the first
overweight person?
Answer:  
1
1

 1.587
p 0.63
(c) What is the probability that an overweight person is found on the 3rd attempt?
P( X  3)  (1  p) n1 p
Answer: P( X  3)  (0.37) 2 (0.63)
P( X  3)  0.0862
(d) What is the probability that it takes more than 5 attempts before an overweight
person is found?
Answer:
P( X  5)  (1  p) n
P( X  5)  (0.37) 5  0.0069
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