Statistics
Midterm Review 2007
Name __________________
Chapter 1
1. The Coca Cola Company has 366,000 stockholders and a poll is conducted by
randomly selecting 30 stockholders from each of the 50 states. The number of
shares held by each sampled stockholder is recorded.
a. Are the values obtained discrete or continuous?
DISCRETE
b. Identify the level of measurement (nominal, ordinal, interval, rational) for
the sample data.
RATIO
c. Which type of sampling (random, systematic, convenience, stratified,
cluster) is being used?
STRATIFIED
d. If the average (mean) number of shares is calculated from the sample
group, is the result a statistic or a parameter?
STATISTIC
e. What is wrong with gauging stockholder views by mailing a questionnaire
that stockholders could complete and mail back?
The sample will be self-selected and thus probably biased.
2. Identify the level of measurement used in each of following.
a. The amounts of tar (in milligrams) in a sample of cigarettes.
RATIO
b. A movie critic’s ratings of “must see, recommended, not recommended,
don’t even think about going.”
ORDINAL
c. A movie critic’s classification of “drama, comedy, adventure”
NOMINAL
3. Determine whether the following values are discrete or continuous.
a. A statistics professor counts 3 absent students.
DISCRETE
b. A statistics professor finds that on the first test, the first paper is turned in
39.627 minutes after the test began.
CONTINUOUS
4. The Hartford Insurance Company has hired you to poll a sample of adults about
their car purchases. What is wrong with using telephone directories as the
population from which the sample is drawn?
It excludes people with unlisted numbers and people without phones
5. Decide whether each description is an observational study or an experiment.
a. Cans of Coke are opened, and the volume (in ounces) of the contents are
measured. Observational study
b. The new drug Statisticzene is tested by recording its effects on the
students who are given the drug. Experiment
c. The effectiveness of multimedia teaching is tested with a sample of
students who complete a course of study using the multimedia approach.
Experiment
d. Much controversy arose over a study of patients with syphilis who were
not given a treatment who could have cured them. Observational study
6. Identify which type of sampling is used: random, systematic, convenience,
stratified, or cluster.
a. The Gallup Organization plans to conduct a poll of New York City
residents with the 212 area code. Computers are used to randomly
generate telephone numbers that are automatically called. Random
b. A marketing expert for MTV is planning a survey in which 500 people
will be randomly selected from each age group of 10-19, 20-29, and so on.
Stratified
c. An ABC news reporter polls people as they pass him on the street.
Convenience
d. A Johns Hopkins University researcher surveys all cardiac patients in each
of 30 randomly selected hospitals.
Cluster
7. Athens Academy is conducting a survey of its students to determine if they prefer
having break before mid-morning meeting or after. It will randomly select 5
students from each advisory to complete the survey.
a. What is the population in this case?
The AA student body
b. What is the sample in this case?
The students selected to take the survey.
c. What type of sampling is AA using?
Stratified.
Chapter 2
Directions: A survey was conducted of men’s heights. The results are summarized
below. Use the frequency table to answer the questions that follow.
Height of Frequency
Men
(inches)
55-59
1
60-64
3
65-69
49
70-74
46
75-79
1
1. Identify the class width, class midpoints, and class boundaries for the frequency
table.
Class width:
5
Class midpoints:
57, 62, 67, 72, 77
Class boundaries:
54.5 – 59.5
59.5 – 64.5
64.5 – 69.5
69.5 – 74.5
74.5 – 79.5
2. Construct a relative frequency table using the data in the frequency table.
55-59
60-64
65-69
70-74
75-79
1%
3%
49%
46%
1%
3. Construct the cumulative frequency table that corresponds to the given frequency
table.
55-59
1
60-64
4
65-69
53
70-74
99
75-79
100
4. Construct a relative frequency histogram using the relative frequency table you
constructed in #2. A relative frequency histogram should have class
boundaries on the x-axis and relative frequencies (percentages) on the y-axis.
The x-axis should be labeled “height (in)” and the y-axis should be labeled
“relative frequency.” The bars of the histogram should be adjacent to one
another with no spaces.
Directions: Fifteen students took an Algebra test. Their grades are listed below. Use
their grades to answer the questions that follow
48
99
75
82
61
44
96
83
92
67
50
75
79
71
91
5. Construct a frequency table for the data given above. You may determine the
class width and the class boundaries.
40-49
2
50-59
1
60-69
2
70-79
4
80-89
2
90-99
4
6. Construct a stem-and-leaf plot for the data above.
4
4 8
5
0
6
7
8
9
1
1
2
1
7
5
3
2
5
9
6
9
8. Construct an ogive for the data given above.
An ogive should have class boundaries on
the x-axis and frequencies on the y-axis.
The x-axis should be labeled “test scores”
and the y-axis should be labeled
“cumulative frequencies.” The ogive
should show cumulative frequencies.
An insurance company is conducting research on how often people see a doctor each
year. Twelve people were surveyed and asked how many times they had seen a doctor in
the past year. Their answers are listed below. Use this data for #9-11
0
5
2
14
12
7
3
2
7
9. Find the mean, median, mode, and midrange of this data.
μ =6
median = 6
mode = 2, 7
10. Find the range and standard deviation of this data.
Range = 14
σ = 4.4
10
1
midrange = 7
9
11. The accompanying frequency table summarizes a sample of human body
temperatures. Find the mean of this frequency table.
μ = 98.17
Temp
0
F
96.5-96.8
Frequency
96.9 – 97.2
8
97.3 – 97.6
14
97.7 – 98.0
22
98.1 – 98.4
19
98.5 – 98.8
32
98.9 – 99.2
6
99.3 – 99.6
4
1
12. If data are normally distributed, what percent of values fall within two standard
deviations of the mean? 95%
13. Heights of women have a normal-distribution with a mean of 63.6 inches and a
standard deviation of 2.5 inches. Using the empirical rule, what is the
approximate percentage of women with heights between
a) 61.1 inches and 66.1 inches? 68%
b) 56.1 inches and 71.1 inches? 99.7%
c) Use the range rule of thumb to identify the lowest and highest unusual heights for
women. Min = 58.6 inches
Max = 68.6 inches
14. What do you know about the values in a data set if the data have a standard deviation
of 0? All the values in the data set are equal.
Directions: For the following data sets, find the mean, standard deviation, and maximum
and minimum usual values. Then, find the z-score of the highest value in the set and the
lowest value in the set.
15. The given values are the measurements of distances (in millimeters) between the
pupils of adult patients being fitted for eyeglasses.
67
66
59
62
x = 64.25 mm
63
66
66
s = 2.7 mm
65
z59 = -1.94
z67 = 1.02
(Remember to use x and s when you are determining the mean and standard
deviation of a sample rather than a population. This is a sample because it does not
include all adult eyeglass wearers.) *Note that this is different than what the
original, handwritten answer key said. The same applies for #16.
16. The given values are the service times (in seconds) of McDonald’s drive-through
customers.
88
107
35
93
x = 81 sec
65
55
s = 26.4 sec
119
83
z35 = -1.74
99
74
46
108
z119 = 1.44
17. What is the z-score of the lowest “usual” value in any data set (according to the
range rule of thumb)? -2
18. What is the z-score of the highest “usual” value in any data set(according to the
range rule of thumb)? 2
19. Would it be unusual for a customer who went to the McDonald’s mentioned in
#16 to wait for only 35 seconds? Why or why not? No, because 35 seconds is
within two standard deviations of the mean.
20. Look at Table 2-16 on page 87 of your book. What percentile is the weight
.8126?
17th percentile
21. Look at Table 2-16 on page 87 of your book. Find the value that marks P60, the
60th percentile. .8189
Chapter 3
Directions: Which method of determining probability is described in each situation
below?
1. Based on today’s temperature, the location of two cold fronts, and the current air
pressure, a meteorologist says there is a 30% chance of rain tomorrow. Subjective
2. Victor Niederhoffer is a professional investor. He tries to predict what the stock
market will do and he buys and sells stock accordingly. In the following excerpt from an
article in the New Yorker, Niederhoffer notices that, for the past three days, the stock
market has opened down.
“Niederhoffer turned to Alex Castaldo, a thin, bespectacled fifty-three-year-old Italian
who has a degree in electrical engineering from M.I.T. and a Ph.D. in finance from
CUNY, and asked him to compile some data. “Doc,” he said to Castaldo, “what does the
market do when it opens down a lot three days in a row?” A few minutes later, Castaldo
handed Niederhoffer a computer printout, which showed that since the start of 2003 there
had been just ten occasions on which, for three consecutive days, the S. & P. 500 had
fallen sharply in the first hour and a half of trading. On eight of those occasions, stocks
had bounced back.”
What method of determining probability is Niederhoffer using to predict what the market
will do? Relative frequency approximation
3. The probability of rolling a three is 1/6. Classical approach
4. If a person is randomly selected, find the probability that his or her birthday is October
18th, which is National Statistics Day in Japan. Ignore leap years.
1
365
5. If a person is randomly selected, find the probability that he or she was NOT born on
National Statistics Day. Ignore leap years.
364
365
Directions: Use the diagram below to answer questions…
If one of the shapes above is selected at random, what is the probability that..
6. it will be a rectangle?
6
14
7. it will have exactly three sides?
8
14
9
14
8. it will have four sides or have the letter B?
9. it will be a have either three or four sides? 1
10. it will have five sides or the letter A?
7
14
11. it will have three sides or the letter A?
10
14
If two of the shapes above are selected at random, with replacement, what is the
probability that…
12. the first shape will be a rectangle and the second will be a triangle?
13. the first will have the letter B and the second will be a rectangle?
48
196
42
196
14. they will both have the letter B?
15. neither will have the letter B?
49
196
49
196
16. the first will be a rectangle or have the letter A and the second will have the letter A
22
and be a rectangle?
196
If two of the shapes above are selected at random without replacement, what is the
probability that…
17. they will both have four sides?
30
182
18. the first will have an A and the second will have a B?
19. they will both be triangles with As?
49
182
20
182
If one shape is picked at random…
20. Are the following events mutually exclusive?
Picking a triangle
Picking a shape with an A
No, because there are triangles with As.
21. Are the following events mutually exclusive?
Picking a square
Picking a triangle?
Yes, no shape is both a triangle and a rectangle.
22. If one shape is picked at random, find the probability that it is a rectangle given that it
4
has a B.
7
23. If one shape is picked at random, find the probability that has an A on it, given that it
2
is rectangle.
6
24. If one shape is picked at random, find the probability that it is a triangle, given that it
is a rectangle. 0
If three shapes are picked at random, with replacement, find the probability that..
25. none of the shapes is a triangle.
216
2744
26. at least one of them is a triangle 1 
27. at least one of them has a B.
216 2528

2744 2744
2401
2744
28. If P(A) = .34, find P(the complement of A) .66
29. Evaluate 5! 120
30. How would you evaluate aPb ?
31. How would you evaluate aCb ?
a!
(a  b)!
a!
(a  b)!b !
32. A small class of five students is instructed to form a straight line. In how many
possible orders can the children line up?
5! = 120
33. A class of ten children is putting on a play. There are four speaking roles. How
many different assignments of speaking roles are there?
10P4 = 5040
34. A class of ten children is putting on a concert. During one song, there is a section of
music that only requires four singers. How many different groups of four can be
selected?
10C4 = 210
35. You have just started your own airline company called Air America. You have one
plane for a route connecting Austin, Boise, and Chicago. How many routes are possible?
3!= 6
35. Walton Pharmaceuticals wants to test the effectiveness of a new drug designed to
relieve allergy symptoms. The initial test will be conducted by treating six people chosen
from a pool of 15 volunteers. If the treatment group is randomly selected, what is the
probability that it consists of the six youngest people in the pool?
1
1

5005
15 C6
36. Each social security number is a sequence of nine digits. If you randomly generate
nine digits, what is the probability of getting your own social security number?
1
1

9
(10) 1, 000, 000, 000
Chapter 4
Directions: Determine if a probability distribution is given. If it is, find μ and σ. If it
isn’t a probability distribution, explain why not.
1.
x
P(x)
0
.125
1
.375
2
.375
3
.125
2.
x
0
1
2
3
4
P(x)
.522
.368
.098
.011
.001
Yes, it is a probability distribution.
μ = 1.5
σ = .9
Yes, it is a probability distribution.
μ = .6
σ = .7
3. Focus groups are often used for finding detailed information about a product. A focus
group of 12 people is randomly selected to discuss products of the Coca Cola Company.
In such groups of 12 people, the mean number who recognize the coca Cola brand name
is 11.4 and the standard deviation is .75. Would it be unusual to randomly select 12
people and find that fewer than eight of them recognize the Coca Cola brand name?
Yes, because 8 is more than two standard deviations below the mean.
4. Look at Table 4-1 on page 181 in your book. Find the probability of getting 10 or
more girls among 14 births. If you are testing a gender-selection technique and you get
10 girls among 14 births, does it appear that the method is effective? Why or why not?
P(10 or more) = .09
Since the probability of getting 10 or more girls naturally is greater than .05, the
technique does not seem to be effective.
5. A scratch-off lottery ticket costs $3.00. Out of every 10,000 scratch-off tickets, 10 of
them pay out a prize of 50, and 2 of them pay out a prize of $1,000. What is the expected
value of one ticket?
Possible Outcome
Win $50
Win $1,000
Lose
X (Value for that
outcome)
$47
$997
-$3
P(x)
.001
.0002
.9988
Expected value = μ = -$2.75
6. What four conditions must a procedure meet in order to yield a binomial probability
distribution?
1. There must be exactly two possible outcomes.
2. There must be a fixed number of trials (n).
3. The events must be independent.
4. The probabilities of success and failure must remain constant.
7. Bryan flips a coin 10 times and records the number of times heads comes up. Does
this yield a binomial probability distribution or not? Why or why not? If so, define S, F,
n, p, and q. Then find P(3) and P(8 or more).
Yes, it does yield a binomial probability distribution.
1. Two outcomes – heads or tails
2. n = 10
3. Independent – one coin flip does not affect the next.
4. The probability of getting heads is always ½ .
S= heads
F = tails
n = 10
p = .5
q = .5
P(3) = .117
P(8 or more) = P(8) + P(9) +P(10) = .055
8. You are testing telephone directory assistance. You request 10 numbers. Assume that
director assistance gives out the wrong number 15% of the time.
a) Find the probability of getting one wrong number.
This is a binomial probability distribution.
1. Two outcomes – correct telephone number or incorrect telephone number.
2. n =10
3. Getting a wrong number one time does not affect the next number you
receive from directory assistance.
4. The probability of getting a wrong number is always .15.
S = getting the wrong number
p = .15
q = .85
F = getting the right number
n = 10
P(1) = .347
b) Find the probability of getting at most one wrong number. P(at most 1) =
P(0) + P(1) = .544
c) If you do get at most one wrong number, does it appear that the rate of wrong
numbers is not 15%?
No. Given a wrong # rate of 15%, it is very likely (> .05) that you will get
either 0 or 1 wrong numbers when you request 10 numbers.
9. Several students are unprepared for a true/false test with 20 questions and all of their
answers are guesses.
a) Find the mean and standard deviation for the number of correct answers for such
students.(Hint: Determine whether or not this is a binomial probability distribution. If it
is, explain why, define S, F, n, x, p, and q and then use the shortcut formulas for μ and
σ.)
This is a binomial distribution.
1. Two outcomes – choose the right answer or the wrong answer
2. n = 20
3. If you are guessing, getting one question right does not affect whether you get
the next question correct.
4. If you are guessing on a true/false test, the probability of choosing the right
answer is always ½ .
S = Choosing the right answer
F = Choosing the wrong answer
n = 20
p = .50
q = .50
When you have a binomial probability distribution, μ = np = 20(.5) = 10
When you have a binomial probability distribution, σ = npq = 2.24
b) Would it be unusual for a student to pass by guessing and getting at least 12 corect
answers? Why or why not? No, because 12 is within two standard deviations of the
mean. Or, No, because P(12 or more) is > .05
10. There are 15 particularly winding miles along Highway 1 where 30 accidents
happened last year.
a. If I randomly select one 1-mile stretch from that treacherous section, what
is the probability that 4 accidents happened in that 1-mile stretch? (Round
answers to three decimal places.)
Use the Poisson distribution here. Interval: 1-mile stretch of highway
Event: accident
30accidents
μ=
= 2 accidents/section
15sec tions
x=4
P(4) = .09
b. How many 1-mile sections would you expect to have been the site of 0
accidents? 1 accident? 2 accidents? 3 accidents? (Round probabilities to
three decimal places and round expected number of sections to one
decimal place.)
# of
# of
accidents sections
0
2.0
1
4.1
2
4.1
3
2.7
Chapter 5
1. The readings on thermometers are normally distributed with μ = 00 C and σ = 10 C.
a. If you randomly select one thermometer, find the probability that its reading is
between .5 and 1.3. .2117
b. If you randomly select one thermometer, find the probability that its reading is
between -.4 and 2.0. .6326
c. If you randomly select one thermometer, find the probability that its reading is
less than -1.7.
.0446
d. Find the temperature that separates the top 7% of thermometers from the rest.
1.480
e. Find the temperature that separates the bottom 11% of thermometers from the
rest.
-1.230
f. Quality control will eliminate the thermometers in the top and bottom 10% . Find
the temperatures that separate the good thermometers from the bad.
-1.280 and 1.280
g. If thermometers must have readings between -1.00 C and 1.00 C to pass quality
control inspections, what percentage of thermometers is discarded for failing
inspection?
31.74%
2. Women’s weights are normally distributed with a mean of 143 lbs and a standard
deviation of 29 lbs.
If one woman is randomly selected, find the probability that her weight is…
a) greater than 158 lbs.
. 3015
b) between 120 lbs and 150 lbs. .4469
c) less than 125 lbs.
.2676
Find the weight that separates…
a) the bottom 15% of weights from the rest of the weights.
112.84 pounds
b) the top 22% of weights from the rest.
159. 82 pounds
If a group of 25 women is randomly selected, find the probability that…
a) the average of their weights is greater than 170.
.0001
b) the average of their weights is between 128 and 149.
.8484