ST 511
MIDTERM EXAM PRACTICE PROBLEMS
Reiland
Material covered on exam:
 all topics covered to this point in the course including binomial, geometric, and
Poisson distributions
 webassign homework #1 through #4
Important Note: This sample exam does not necessarily have a question for every topic that is included on the
midterm exam.
1.
The heights of American men aged 18 to 24 are approximately symmetrically mound shaped with
mean 68 inches and standard deviation 2.5 inches. Half of all young men are shorter than what height?
2.
Use the information in Problem 1. Use the 68-95-99.7 rule to determine the percentage of young men that
are taller than 6' 1" ?
3.
The grade point averages (GPA) of 7 randomly chosen students in a statistics class are
3.14 2.37 2.94 3.60 1.70 4.00 1.85
If (x  x)# œ 4.51, what is the standard deviation?
4.
A survey conducted in a college stats class asked students about the number of credit hours they were taking
that semester. The number of credit hours for a random sample of 16 students is:
10 10 12 14 15 15 15 15
17 17 19 20 20 20 20 22
a. find the mean and standard deviation for the number of credit hours.
b. find the median and IQR for the number of credit hours.
c. is it more appropriate to use the mean and standard deviation or the median and IQR to summarize these
data?
5.
A standardized test designed to measure math anxiety has a mean of 100 and a standard deviation
of 10 in the population of first year college students. Which of the following observations would
you suspect is an outlier?
a) 150 b) 100 c) 90 d) 125 e) none of the above
6.
Twenty-eight applicants interested in working for the Food Stamp program took an examination designed to
measure their aptitude for social work. The following test scores were obtained:
79, 93, 84, 86, 77, 63, 46, 97, 87, 88, 87, 92, 68, 72, 86, 98, 81, 70, 66, 98, 59, 76, 68, 91, 94,
85, 88, 79.
a. Find U" , the first quartile.
b. Construct a boxplot for these observations. Do you observe any outliers?
7.
The boxplots show the age of people involved in traffic-related accidents according to their role in the
accident.
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Midterm Exam Practice Problems
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a. Which role involved the youngest person, and what is the age?
,
b. Which role had the lowest median, and what is the age?
,
c. Which role had the smallest range of ages, and what is it?
,
d. Which role had the largest IQR of ages, and what is it?
,
e. Which role generally had the oldest people?
8.
A manufacturer of television sets has found that for the sets he produces, the lengths of time until the first
repair can be described using a normal model with a mean of 4.5 years and a standard deviation of 1.5 years.
If the manufacturer sets the warrantee so that only 10.2% of the 1st repairs are covered by the warrantee,
how long should the warrantee last?
9.
Suppose the amount of tar in cigarettes can be described using a normal model with a mean of 3.5 mg and a
standard deviation of 0.5 mg.
a. What proportion of cigarettes have a tar content that exceeds 4.25 mg?
b. In order to advertise as a low tar brand, a manufacturer must prove that their tar content is below the 25th
percentile of the tar content distribution. Find the 25th percentile of the distribution of tar amounts.
10. The mean SAT verbal score of next year's freshmen entering the local university is 600. It is also known
that 69.5% of these freshmen have scores that are less than 625. If the scores can be described using a
normal model, what is the standard deviation of the scores?
11. Two students are enrolled in an introductory statistics course at the University of Florida. The first student
is in a morning section and the second student is in an afternoon section. If the student in the morning
section takes a midterm and earns a score of 76, while the student in the afternoon section takes a midterm
with a score of 72, which student has performed better compared to the rest of the students in his respective
class? Assume that the test scores follow a normal model. For the morning class, the class mean was 64
with a standard deviation of 8. For the afternoon class, the class mean was 60 with a standard deviation of
7.5.
12. The following data are a sample of the percentage rates of return experienced by six-month Eurodollar
deposit purchasers:
1
2 2 1 1 4 2 3 1
ST 511
Midterm Exam Practice Problems
The standard deviation is:
a. 4 b. 2 c. 1.05 d. 1.11
page 3
e. can't be done since some values are negative
13. The highway gas mileage B, measured in miles per gallon (mpg), of 26 models of midsize cars, have the
following summary statistics: B œ 26.54 mpg, median œ 26 mpg, = œ 3.04 mpg, IQR œ 3 mpg. If you
convert gas mileage B from miles per gallon to B8/A which is measured in miles per liter, what are the new
values of the summary statistics? (3.785 liters œ 1 gallon).
14. Shown below is the normal probability plot for 200 monthly telephone bills.
Norm al Quantile Plot
140
phone bills
120
100
80
60
40
20
0
-4
-3
-2
-1
0
1
2
3
4
z-score
Shown below is a histogram. Is this a histogram of the same data that was used to construct the normal
probability plot?
Histogram
Frequency
40
30
20
10
0
15. A local plumber makes house calls. She charges $30 to come out to the house and $40 per hour for her
services. For example, a 4-hour service call costs $30 + $40(4) = $190.
a. The table shows summary statistics for the length of service calls for the past month. Fill in the table to
find out the cost of the service calls.
Statistic Hours of Service Call Cost of Service Call
Mean
4.5
Median
3.5
Stan Dev
1.2
IQR
2.0
Minimum
0.5
b. This past month, the time the plumber spent on a particular service call had a z-score of  1.50. What
is the z-score for the cost of the service call?
16. In 2000 the Department of Education published the Digest for Education Statistics, a collection of
information about education in the United States. They reported the average amount (dollars per student)
ST 511
Midterm Exam Practice Problems
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spent by public schools in each state and Washington, D.C.during the school year 1997-8. The data was
recorded according to whether the state lies east or west of the Mississippi River. A back-to-back stem and
leaf display of the data is shown below. 6|7 denotes $6,700.
a. Which states, Eastern or Western, tend to spend more?
b. Western states median = ? Eastern states Q" = ?
17. There are 25 pies entered in the pie-baking contest at the state fair. How many ways can the judges select 3
pies to finish 1st, 2nd, and 3rd?
18. Four radar systems are arranged so that they work independently of each other. Each system has a 0.9
chance of detecting an approaching airborne object. Find the probability that at least one radar system will
fail to detect an approaching object.
19. Let \ be the number of accidents per week at a hazardous intersection; \ varies with mean 2.2 and
standard deviation 1.4. Let \" ß \# , and \$ be the number of accidents in each of 3 different weeks at this
intersection. What is 5Ð\" \# \$ Ñ , the standard deviation of the sum \"  \#  \$ ?
20. Determine which of the following functions is in fact a probability distribution function.
a.
p(x) œ
1
4,
b.
p(x) œ
x#
25 ,
c.
p(x) œ
5x#
6 ,
x œ 3, 4, 5, 6.
x œ 0, 1, 2, 3, 4.
x œ 0, 1, 2, 3.
21. Assume that 70% of teenagers who go to take the written drivers license test have studied for the test. Of
those who study for the test, 95% pass; of those who do not study for the test, 60% pass. What is the
probability that a teenager who passes the written drivers license test did not study for the test?
22. A friend of yours plans to toss a fair coin 200 times. You watch the first 20 tosses and are surprised that she
got 15 heads. But then you get bored and leave. How many heads do you expect her to have when she has
finished all 200 tosses?
ST 511
Midterm Exam Practice Problems
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23. Assume the heights of high school basketball players can be described with a normal model. For boys the
mean is 74 inches with a standard deviation of 4.5 inches, while girl players have a mean height of 70 inches
and standard deviation 3 inches. At a mixed 2-on-2 tournament teams are formed by randomly pairing boys
with girls as teammates.
a. On average, how much taller do you expect the boy to be?
b. What will be the standard deviation of the difference in teammates’ heights?
c. On what fraction of the teams would you expect the girl to be taller than the boy?
24. A certain hotel chain finds that 20% of the reservations made will not be used.
a. What is the probability that among fifteen reservations made that more than 8 but less than 12 will be
used?
b. If four reservations are made, what is the probability that less than two will be unused?
25. The taste test for PTC (phenylthiourea) is a favorite exercise for every human genetics class. It has been
established that a single gene determines the characteristic, and that 70% of the American population are
"tasters," while 30% are "non-tasters". If 20 people are randomly chosen and administered the test:
a. give the probability distribution of x, the number of "non-tasters" out of the 20 chosen.
b. Find P(3 <x < 9).
c. Find the mean of x.
d. Find the variance of x.
26. During the lunch-hour rush at a McDougalds restaurant orders for the Big Mac hamburger follow a Poisson
distribution and occur at a rate of 4 per minute. What is the probability that 5 or more Big Macs will be
ordered in one minute?
27. The owner of a small convenience store notices that only 5% of customers buy magazines.
a. What is the probability that the first customer to buy a magazine is the 4>2 customer?
b. What is the probability that the first customer to buy a magazine is the 8>2 customer?
c. How many customers should the owner expect until a customer buys a magazine?
28. The colors of M&M candies in a typical bag have the following probabilities: P(brown) œ !Þ$!,
P(red) œ !Þ#!, P(yellow) œ !Þ#!, P(green) œ !Þ"!, P(orange) œ !Þ"!, P(blue) œ !Þ"!. You have just
purchased a bag of M&M's and select one candy at a time from the bag. What is the probability that the first
red one is the 4th candy you select from the bag?
29. What is the expected number of tosses of a fair six-sided die until each of the six numbers appears at least
once?
SOLUTIONS
1. 68 inches, the mean 2. 2.5% 3. .87 4. a. B œ "'Þ$ credit hours, = œ $Þ( credit hours; b. median = 16
credit hours, IQR = Q$  Q" œ #!  "%Þ& œ &Þ& credit hours; c. the histogram is shown below; since the data
are not symmetric and mound-shaped, it is more appropriate to use the median and IQR to summarize the data
ST 511
Midterm Exam Practice Problems
page 6
5. a. 150
6. a. The first step is to order the data. Then compute the overall median. Since there are 28 observations,
the overall median is the mean of the 2 middle observations in positions 14 and 15. Median
œ )%)&
œ 84.5. Compute Q" : we want the median of the lower half of the data (since there an even
#
number of observations, do not include the overall median in either half). The median of the smallest 14
observations is the mean of the 2 middle observations in positions 7 and 8, so Q" œ (!(#
œ (".
#
b.
Test Scores
0
20
40
60
80
100
120
No outliers are observed.
7. a. passenger, less than 1 year; b. passenger, 21 years; c. cyclist, 40 years; d. pedestrian, 44 years; e.
Pedestrian; while the oldest person involved in an accident is not a pedestrian, the median for pedestrians is
almost 45 years, while the median age in the other groups is between 22 and 35 years old. The oldest 50% of the
pedestrian group, from 45 to 87, is generally older than the youngest 75% of two groups - cyclist and passenger,
and only the driver group has any of its middle 50% as old. The driver and passenger groups have a few people
older than the pedestrian group.
8. z = -1.27 ; x = 2.595 years. 9. a. .0668 b. z = -0.675 ; Q" = 3.16. 10. 0.51 = (625-600)/5 Ê 5 œ 49.02
11. z1 =(76-64)/8=1.5 ; z2 =(72-60)/7.5=1.6 . The student in the afternoon section performed better.
12. b.
13. B8/A œ 7.01 miles per liter; median8/A œ 6.87 miles per liter; =8/A œ .803 miles per liter; MUV8/A œ Þ793
miles per liter. 14. Yes. Note upward curvature (or “leveling off”) and high density of points in left portion of
probability plot (no left tail since bills are not less than $0); plot very “steep” in central portion and somewhat
sparse, which means there are not many observations in middle of data (no high point in middle of histogram);
downward curvature in right portion (to the right of D = 1) of plot indicates the higher phone bills but no points
that you would consider to be outliers.
15. a.
Statistic Hours of Service Call Cost of Service Call
Mean
4.5
$210
Median
3.5
$170
Stan Dev
1.2
$48
IQR
2.0
$80
Minimum
0.5
$50
b.  1.50
16. a. Eastern b. West has 24 data values; West median = mean of data values in ordered positions 12 and 13:
West median œ $&ß*!!# $'ß!!! œ $5ß 950. East has 27 data values; East median is data value in ordered postion
14, so East median is $6,800 (last 8 in stem row 6). East Q" is the median of the lower half; since there are an
odd number of East data values, the overall East median (last 8 in stem row 6) is included in both halves when
determining the quartiles. East Q" is the median of the smallest 14 data values, so East Q" = the mean of the
numbers in ordered positions 7 and 8 = $&ß*!!# $'ß"!! œ $'ß !!!. 17. #& T$ œ "$ß )!!. 18. The probability that no
radar system fails to detect an airborne object (i. e. all four radar systems work) is (.9)% , therefore the probability
that at least one fails is 1  (.9)% . 19. Z +<Ð\"  \#  \$ Ñ œ Z +<(X" Ñ  Z +<Ð\# Ñ  Z +<Ð\$ Ñ œ 5.88; so
ST 511
Midterm Exam Practice Problems
page 7
5Ð\" \# \$ Ñ œ &Þ)) œ 2.425. 20. a. is the only legitimate prob. distribution since in b. p(x)  1; in c.
all x
p(3)  0.
21.
T Ð:+==Ñ œ !Þ''&  !Þ")! œ ! Þ)%&Þ Given that a student passed, the probability the student did not study is
!Þ")!
!Þ")!
!Þ''&!Þ")! œ !Þ)%& œ !Þ#"$.
22. 105 23. a. Let B  G denote the boy's height minus the girl's height; E(B – G) = E(B) – E(G) = 74 – 70 = 4
inches; b. Var(B – G) = Var(B) + Var(G) = (4.5)# + (3)# = 29.25. SD(B – G) = 29.25 = 5.4; c. if the girl is

taller than the boy, then B  G  !. T ÐF  K  !Ñ œ T  FK%
 !%
&Þ%
&Þ% œ T ÐD   !Þ(%Ñ. From the
standard normal table the proportion of the area to the left of  !Þ(% is !Þ##*'Þ
24Þ a. p = 0.8; P(8 < x < 12) = 0.334 b. p = 0.2; P(x < 2) = 0.8192
20!
25. a. p(x) = x!(20-x)!
(0.3)x (0.7)20-x x = 0,1,2,...,20 b. 0.78 c. 6 d. 4.2
26. T Ð\ &Ñ œ "  T Ð\ Ÿ %Ñ œ "  Þ'#)) œ Þ$("#
"
27. geometric. a. :Ð%Ñ œ Ð!Þ*&Ñ$ Ð!Þ!&Ñ œ !Þ!%$; b. :Ð)Ñ œ Ð!Þ*&Ñ( Ð!Þ!&Ñ œ !Þ!$%*; c. IÐ\Ñ œ Þ!&
œ #!
28. geometric; success = red candy; P(success) œ !Þ#!;
T Ð0 3<=> </. 3= %>2 -+8.C =/6/->/.Ñ œ Ð!Þ)!Ñ$ Ð!Þ#!Ñ œ !Þ"!#%
29. "  "&  "%  "$  "#  "" œ "  '&  '%  '$  '#  '" œ "%Þ(
'
'
'
'
'
Use the expected value of the geometric random variable.The first roll of a die necessarily gives us a number not
rolled before. After we've rolled this first number (whatever it is), the probability we roll a different number is
5/6 since we want to roll one of the 5 numbers we haven't yet rolled. The expected number of additional rolls
until a second number turns up is thus 1/(5/6) or 6/5. After rolling two different numbers, the probability we roll
a third number different from the first two is 4/6, and so the expected number of additional rolls until it turns up
is 1/(4/6) or 6/4. Continue in this manner to obtain the answer shown above.