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Fundamentals of Statistics

Michael Sullivan III

Chapter 3

Numerically Summarizing Data - all with Video Answers

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Section 1

Measures of Central Tendency

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Problem 1

Concepts and Vocabulary
What does it mean if a statistic is resistant? Why is the median resistant, but the mean is not? Is the mode a resistant measure of center?

Emily Himsel
Emily Himsel
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01:26

Problem 2

Describe how the mean and the median can be used to determine the shape of a distribution.

Manisha Sarker
Manisha Sarker
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05:32

Problem 3

In the 2000 census conducted by the U.S. Census Bureau, two average household incomes were reported: $\$ 41,349$ and $\$ 55,263 .$ One of these averages is the mean and the other is the median. Which is the mean? Support your answer.

Jon Southam
Jon Southam
Numerade Educator
00:56

Problem 4

The U.S. Department of Housing and Urban Development (HUD) uses the median to report the average price of a home in the United States. Why do you think HUD uses the median?

Trinity Steen
Trinity Steen
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01:21

Problem 5

A histogram of a set of data indicates that the distribution of the data is skewed right. Which measure of central tendency will be larger, the mean or the median? Why?

Manisha Sarker
Manisha Sarker
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01:05

Problem 6

If a data set contains 10,000 values arranged in increasing order, where is the median located?

Narayan Hari
Narayan Hari
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01:40

Problem 7

Explain why the mode is used as the measure of central tendency for qualitative data.

Manisha Sarker
Manisha Sarker
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01:12

Problem 8

$\mathbf{A}(\mathbf{n})$_______is a descriptive measure of a population, and $a(n)$_______is a descriptive measure of a sample.

Manisha Sarker
Manisha Sarker
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01:16

Problem 9

True or False: A data set will always have exactly one mode.

Narayan Hari
Narayan Hari
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01:16

Problem 10

True or False: If the number of observations is odd, the median is $M=\frac{n+1}{2}.$

Manisha Sarker
Manisha Sarker
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01:05

Problem 11

Find the population mean or sample mean as indicated.
Sample: 20,13,4,8,10

Manisha Sarker
Manisha Sarker
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01:09

Problem 12

Find the population mean or sample mean as indicated.
Sample: 83,65,91,87,84

Manisha Sarker
Manisha Sarker
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01:10

Problem 13

Find the population mean or sample mean as indicated.
Population: 3,6,10,12,14

Manisha Sarker
Manisha Sarker
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01:16

Problem 14

Find the population mean or sample mean as indicated.
Population: 1,19,25,15,12,16,28,13,6

Manisha Sarker
Manisha Sarker
Numerade Educator
01:00

Problem 15

For Super Bowl XXXIX, Fox television sold 59 ad slots for a total revenue of roughly $\$ 142$ million. What was the mean price per ad slot?

Nick Johnson
Nick Johnson
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01:07

Problem 16

The median for the given set of six ordered data values is $26.5 .$ What is the missing value? 71221 ______ 41 50.

Jerelyn Nevil
Jerelyn Nevil
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02:21

Problem 17

The Insurance Institute for Highway Safety crashed the 2001 Honda Civic four times at 5 miles per hour. The costs of repair for each of the four crashes were
$$
\$ 420, \$ 462, \$ 409, \$ 236
$$
Compute the mean, median, and mode cost of repair.

Manisha Sarker
Manisha Sarker
Numerade Educator
01:45

Problem 18

Cell Phone Use The following data represent the monthly cell phone bill for my wife's phone for six randomly selected months.
$$
\$ 35.34, \$ 42.09, \$ 39.43, \$ 38.93 \$ 43.39, \$ 49.26
$$
Compute the mean, median, and mode phone bill.

Trinity Steen
Trinity Steen
Numerade Educator
02:39

Problem 19

Concrete Mix A certain type of concrete mix is designed to withstand 3000 pounds per square inch (psi) of pressure. The strength of concrete is measured by pouring the mix into casting cylinders 6 inches in diameter and 12 inches tall. The cylinder is allowed to "set up" for 28 days.The cylinders are then stacked on one another until the cylinders are crushed. The following data represent the strength of nine randomly selected casts (in psi). 3960,4090,3200,3100,2940,3830,4090,4040,3780
Compute the mean, median, and mode strength of the concrete (in psi).

Manisha Sarker
Manisha Sarker
Numerade Educator
02:17

Problem 20

Flight Time The following data represent the flight time (in minutes) of a random sample of seven flights from Las Vegas, Nevada, to Newark, New Jersey, on Continental Airlines.
$$
282,270,260,266,257,260,267
$$
Compute the mean, median, and mode flight time.

Narayan Hari
Narayan Hari
Numerade Educator
01:26

Problem 21

Compute the mean, median, and mode flight time. For each of the three histograms shown, determine whether the mean is greater than, less than, or approximately equal to the median. Justify your answer.
(FIGURE CANNOT COPY)

Manisha Sarker
Manisha Sarker
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03:48

Problem 22

Match the histograms shown to the summary statistics shown to the right:
(GRAPH CAN'T COPY)

Jon Southam
Jon Southam
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05:43

Problem 23

Applying the concepts
ATM Fees The following data for a random sample of banks in Los Angeles and New York City represent the ATM fee for using another bank's ATM.

Barsha Rana
Barsha Rana
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09:58

Problem 24

In an experiment conducted online at the University of Mississippi, study participants are asked to react to a stimulus. In one experiment, the participant must press a key upon seeing a blue screen. The time (in seconds) to press the key is measured. The same person is then asked to press a key upon seeing a red screen, again with the time to react measured. The table shows the results for six study participants. Compute the mean, median, and mode reaction time for both blue and red. Does there appear to be a difference in the reaction time? What might account for any difference? How might this information be used?

Jon Southam
Jon Southam
Numerade Educator
11:40

Problem 25

The following data represent the pulse rates (beats per minute) of nine students enrolled in a section of Sullivan's Introductory Statistics course. Treat the nine students as a population.
(a) Compute the population mean pulse.
(b) Determine two simple random samples of size 3 and compute the sample mean pulse of each sample.
(c) Which samples result in a sample mean that overestimates the population mean? Which samples result in a sample mean that underestimates the population mean? Do any samples lead to a sample mean that equals the population mean?

Jon Southam
Jon Southam
Numerade Educator
11:26

Problem 26

The following data represent the travel time (in minutes) to school for nine students enrolled in Sullivan's College Algebra course. Treat the nine students as a population.
(a) Compute the population mean for travel time.
(b) Determine three simple random samples of size 4 and compute the sample mean for travel time of each sample.
(c) Which samples result in a sample mean that overestimates the population mean? Which samples result in a sample mean that underestimates the population mean? Do any samples lead to a sample mean that equals the population mean?

Jon Southam
Jon Southam
Numerade Educator
00:41

Problem 27

Mia Hamm, who retired after the 2004 Olympics, is considered by some to be the most prolific player in international soccer. The following data represent the number of goals scored over her 18 -year career.
(a) Compute the population mean of the number of goals she scored.
(b) Determine two simple random samples of size 3 and compute the sample mean of the number of goals she scored.
(c) Which samples result in a sample mean that overestimates the population mean? Which samples result in a sample mean that underestimates the population mean? Do any samples lead to a sample mean that equals the population mean?

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
07:17

Problem 28

Lance Armstrong won the Tour de France seven consecutive times $(1999-2005) .$ The following table gives the winning times, distances, speeds, and margin of victory.
(a) Compute the mean and median of his winning times for the six races.
(b) Compute the mean and median of the distances for the six races.
(c) Compute the mean and median of his winning time margins.
(d) Compute the mean winning speed by finding the mean of the data values in the table. Next, compute the mean winning speed by finding the total of the six distances and dividing by the total of the six winning times. Finally, compute the mean winning speed by dividing the mean distance by the mean winning time. Do the three values agree or are there differences?

Jon Southam
Jon Southam
Numerade Educator
00:23

Problem 29

Connection Time The following data represent the connection time in seconds to an Internet service provider for 30 randomly selected connections.
$$\begin{array}{llllll}
39.76 & 36.13 & 36.61 & 38.80 & 39.04 & 39.09 \\
\hline 37.24 & 35.62 & 40.07 & 38.76 & 39.23 & 38.38 \\
\hline 38.24 & 36.34 & 35.89 & 42.86 & 36.03 & 37.03 \\
\hline 38.64 & 41.86 & 41.22 & 37.19 & 40.50 & 39.81 \\
\hline 39.84 & 39.45 & 40.91 & 43.12 & 40.54 & 42.02 \\
\hline
\end{array}$$
A histogram of the data is shown. The mean connection time is 39.007 seconds and the median connection time is 39.065 seconds. Use this information to identify the shape of the distribution. Which measure of central tendency better describes the "center" of the distribution?

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
00:29

Problem 30

The following data represent the annual subscription cost (in dollars) for a random sample of 26 biology journals.
$$\begin{array}{llllll}
\hline 1188 & 778 & 1970 & 661 & 1294 & 1175 \\
\hline 2033 & 3911 & 198 & 8415 & 796 & 1840 \\
\hline 1141 & 1050 & 3643 & 1407 & 1092 & 585 \\
\hline 1049 & 1092 & 1589 & 4115 & 1150 & 2799 \\
\hline 707 & 2330 & & & & \\
\hline
\end{array}$$
A histogram of the data is shown. The mean subscription cost is $\$ 1846$ and the median subscription cost is $\$ 1182 .$ Use this information to identify the shape of the distribution. Which measure of central tendency better describes the "center" of the distribution?

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
00:37

Problem 31

Dr. Paul Oswiecmiski randomly selects 40 of his $20-$ to 29 -year-old patients and obtains the following data regarding their serum HDL cholesterol.
$$\begin{array}{llllllll}
70 & 56 & 48 & 48 & 53 & 52 & 66 & 48 \\
\hline 36 & 49 & 28 & 35 & 58 & 62 & 45 & 60 \\
\hline 38 & 73 & 45 & 51 & 56 & 51 & 46 & 39 \\
\hline 56 & 32 & 44 & 60 & 51 & 44 & 63 & 50 \\
\hline 46 & 69 & 53 & 70 & 33 & 54 & 55 & 52
\end{array}$$
(a) Compute the mean and the median serum HDL.
(b) Identify the shape of the distribution based on the histogram drawn in Problem 31 in Section 2.2 and the relationship between the mean and the median.

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
00:33

Problem 32

The volume of a stock is the number of shares traded on a given day. The following data represent the volume of Altria Group stock traded for a random sample of 35 trading days in 2004 . The data are in millions, so 3.78 represents 3,780,000 shares traded.
$$\begin{array}{|lllll}
3.78 & 8.74 & 4.35 & 5.02 & 8.40 \\
\hline 6.06 & 5.75 & 5.34 & 6.92 & 6.23 \\
\hline 5.32 & 3.25 & 6.57 & 7.57 & 6.07 \\
\hline 3.04 & 5.64 & 5.00 & 7.16 & 4.88 \\
\hline 10.32 & 3.38 & 7.25 & 6.52 & 4.43 \\
\hline 3.38 & 5.53 & 4.74 & 9.70 & 3.56 \\
\hline 10.96 & 4.50 & 7.97 & 3.01 & 5.58
\end{array}$$
(a) Compute the mean and the median number of shares traded.
(b) Identify the shape of the distribution based on the histogram drawn in Problem 32 in Section 2.2 and the relationship between the mean and the median.

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
00:38

Problem 33

The following data represent the weights (in grams) of a simple random sample of $50 \mathrm{M} \& \mathrm{M}$ plain candies.
$$\begin{array}{lllllll}
0.87 & 0.88 & 0.82 & 0.90 & 0.90 & 0.84 & 0.84 \\
\hline 0.91 & 0.94 & 0.86 & 0.86 & 0.86 & 0.88 & 0.87 \\
\hline 0.89 & 0.91 & 0.86 & 0.87 & 0.93 & 0.88 & \\
\hline 0.83 & 0.94 & 0.87 & 0.93 & 0.91 & 0.85 & \\
\hline 0.91 & 0.91 & 0.86 & 0.89 & 0.87 & 0.93 & \\
\hline 0.88 & 0.88 & 0.89 & 0.79 & 0.82 & 0.83 & \\
\hline 0.90 & 0.88 & 0.84 & 0.93 & 0.76 & 0.90 & \\
\hline 0.88 & 0.92 & 0.85 & 0.79 & 0.84 & 0.86 & \\
\hline
\end{array}$$
Determine the shape of the distribution of weights of M\&Ms by drawing a frequency histogram and computing the mean and median. Which measure of central tendency better describes the weight of a plain M\&M?

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
00:29

Problem 34

Old Faithful We have all heard of the Old Faithful geyser in Yellowstone National Park. However, there is another, less famous, Old Faithful geyser in Calistoga, California. The following data represent the length of eruption (in seconds) for a random sample of eruptions of the California Old Faithful.
$$\begin{array}{lllllll}
108 & 108 & 99 & 105 & 103 & 103 & 94 \\
\hline 102 & 99 & 106 & 90 & 104 & 110 & 110 \\
\hline 103 & 109 & 109 & 111 & 101 & 101 & \\
\hline 110 & 102 & 105 & 110 & 106 & 104 & \\
\hline 104 & 100 & 103 & 102 & 120 & 90 & \\
\hline 113 & 116 & 95 & 105 & 103 & 101 & \\
\hline 100 & 101 & 107 & 110 & 92 & 108 &
\end{array}$$
Determine the shape of the distribution of time between eruptions by drawing a frequency histogram and computing the mean and median. Which measure of central tendency better describes the time between eruptions?

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
00:30

Problem 35

Hours Working A random sample of 25 college students was asked, "How many hours per week typically do you work outside the home?" Their responses were as follows:
$$\begin{array}{ccccc}
0 & 0 & 15 & 20 & 30 \\
\hline 40 & 30 & 20 & 35 & 35 \\
\hline 28 & 15 & 20 & 25 & 25 \\
\hline 30 & 5 & 0 & 30 & 24 \\
\hline 28 & 30 & 35 & 15 & 15
\end{array}$$
Determine the shape of the distribution of hours worked by drawing a frequency histogram and computing the mean and median. Which measure of central tendency better describes hours worked?

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
00:26

Problem 36

A Dealer's Profits The following data represent the profits (in dollars) of a new car dealer for a random sample of 40 sales.
$$\begin{array}{rrrrr}
781 & 1,038 & 453 & 1,446 & 3,082 \\
\hline 501 & 451 & 1,826 & 1,348 & 3,001 \\
\hline 1,342 & 1,889 & 580 & 0 & 2,909 \\
\hline 2,883 & 480 & 1,664 & 1,064 & 2,978 \\
\hline 149 & 1,291 & 507 & 261 & 540 \\
\hline 543 & 87 & 798 & 673 & 2,862 \\
\hline 1,692 & 1,783 & 2,186 & 398 & 526 \\
\hline 730 & 2,324 & 2,823 & 1,676 & 4,148
\end{array}$$
Determine the shape of the distribution of new car profits by drawing a frequency histogram and computing the mean and median. Which measure of central tendency better describes the profit?

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
01:26

Problem 37

The following data represent the region of birth of foreign-born residents of the United States in $2003 .$ Determine the mode region of birth.

Manisha Sarker
Manisha Sarker
Numerade Educator
01:10

Problem 38

Robbery The following data represent the number of offenses for various robberies in $2003 .$ Determine the mode offense.

Manisha Sarker
Manisha Sarker
Numerade Educator
01:23

Problem 39

2004 Presidential Election An exit poll was conducted in Los Alamos County, New Mexico, in which a random sample of 40 voters revealed whom they voted for in the presidential election. The results of the survey are shown below. Determine the mode candidate.
$$\begin{array}{lllll}
\text { Kerry } & \text { Kerry } & \text { Bush } & \text { Bush } & \text { Bush } \\
\hline \text { Bush } & \text { Kerry } & \text { Kerry } & \text { Bush } & \text { Bush } \\
\hline \text { Kerry } & \text { Bush } & \text { Kerry } & \text { Bush } & \text { Kerry } \\
\hline \text { Bush } & \text { Bush } & \text { Kerry } & \text { Kerry } & \text { Nader } \\
\hline \text { Kerry } & \text { Bush } & \text { Bush } & \text { Kerry } & \text { Kerry } \\
\hline \text { Badnarik } & \text { Kerry } & \text { Bush } & \text { Bush } & \text { Bush } \\
\hline \text { Bush } & \text { Bush } & \text { Bush } & \text { Bush } & \text { Kerry } \\
\hline \text { Kerry } & \text { Kerry } & \text { Kerry } & \text { Bush } & \text { Bush }
\end{array}$$

Manisha Sarker
Manisha Sarker
Numerade Educator
00:59

Problem 40

Hospital Admissions The following data represent the diagnosis of a random sample of 20 patients admitted to a hospital. Determine the mode diagnosis.

Nick Johnson
Nick Johnson
Numerade Educator
01:07

Problem 41

Each of the following three data sets represents the IQ scores of a random sample of adults. IQ scores are known to have a mean and median of $100 .$ For each data set, compute the mean and median. For each data set recalculate the mean and median, assuming that the individual whose IQ is 106 is accidentally recorded as $160 .$ For each sample size, state what happens to the mean and the median. Comment on the role the number of observations plays in resistance.

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
06:48

Problem 42

Super Bowl XXXIX Champion New England Patriots The following table gives roster information for the offense of the Super Bowl XXXIX Champion New England Patriots.
(a) Find the mean, median, and mode age.
(b) Find the mean, median, and mode weight.
(c) Find the mean, median, and mode years of experience. (Note: Rookie $=0$ years.)
(d) Find the mode college attended.
(e) Obtain a simple random sample of six members of New England's offense. Compute sample mean age, weight, and years of experience. How do the sample means compare to the population means?
(f) Compute the mean, median, and mode weights of the five offensive guards $(\mathrm{OG}) .$ Compute the mean, median, and mode weights of the five running backs (RB). Does there appear to be a difference in the weights? What might account for any differences?
(g) Does it make sense to compute the mean player number? Why?

Jon Southam
Jon Southam
Numerade Educator
02:04

Problem 43

Using the data presented in Problem $42,$ answer the following.
(a) The skilled positions in football are quarterback (QB), running back (RB), wide receiver (WR), and tight end (TE). Obtain a stratified sample by using skilled positions as one stratum and the remaining positions as a second stratum. Randomly select four skilled players and two "nonskilled" players. Compute the mean weight of the sample data. Compare to the result in part (e) in Problem 42.
(b) Cluster the players by position. Obtain a cluster sample by randomly selecting two clusters. Compute the mean weight of the sample data. Compare to the result in to part (e) in Problem 42. Can you think of any problems with obtaining a cluster sample?

Jon Southam
Jon Southam
Numerade Educator
01:31

Problem 44

You are negotiating a contract for the Players' Association of the NBA. Which measure of central tendency will you use to support your claim that the average player's salary needs to be increased? Why? As the chief negotiator for the owners, which measure would you use to refute the claim made by the Players' Association?

Christopher Stanley
Christopher Stanley
Numerade Educator
00:45

Problem 45

In January $2005,$ the mean amount of money lost per visitor to a local riverboat casino was $\$ 135 .$ Do you think the median was more than, less than, or equal to this amount? Why?

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
02:16

Problem 46

Missing Exam Grade A professor has recorded exam grades for 20 students in his class, but one of the grades is no longer readable. If the mean score on the exam was 82 and the mean of the 19 readable scores is $84,$ what is the value of the unreadable score?

Christopher Stanley
Christopher Stanley
Numerade Educator
02:16

Problem 47

Missing Exam Grade A professor has recorded exam grades for 20 students in his class, but one of the grades is no longer readable. If the mean score on the exam was 82 and the mean of the 19 readable scores is $84,$ what is the value of the unreadable score?

Christopher Stanley
Christopher Stanley
Numerade Educator
01:09

Problem 48

For each of the following situations, determine which measure of central tendency is most appropriate and justify your reasoning.
(a) Average price of a home sold in Pittsburgh, Pennsylvania, in 2002
(b) Most popular major for students enrolled in a statistics course
(c) Average test score when the scores are distributed symmetrically
(d) Average test score when the scores are skewed right
(e) Average income of a player in the National Football League
(f) Most requested song at a radio station

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
06:14

Problem 49

Linear Transformations Benjamin owns a small Internet business. Besides himself, he employs nine other people. The salaries earned by the employees are given below in thousands of dollars (Benjamin's salary is the largest, of course):
$$
30,30,45,50,50,50,55,55,60,75
$$
(a) Determine the mean, median, and mode for salary.
(b) Business has been good! As a result, Benjamin has a total of $\$ 25,000$ in bonus pay to distribute to his employees. One option for distributing bonuses is to give each employee (including himself) $\$ 2500$ Add the bonuses under this plan to the original salaries to create a new data set. Recalculate the mean, median, and mode. How do they compare to the originals?
(c) As a second option, Benjamin can give each employee a bonus of $5 \%$ of his or her original salary. Add the bonuses under this second plan to the original salaries to create a new data set. Recalculate the mean, median, and mode. How do they compare to the originals?
(d) As a third option, Benjamin decides not to give his employees a bonus at all. Instead, he keeps the $\$ 25,000$ for himself. Use this plan to create a new data set. Recalculate the mean, median, and mode. How do they compare to the originals?

Jon Southam
Jon Southam
Numerade Educator
00:55

Problem 50

Use the five test scores of 65,70
$71,75,$ and 95 to answer the following questions:
(a) Find the sample mean.
(b) Find the median.
(c) Which measure of central tendency best describes the typical test score?
(d) Suppose the professor decides to curve the exam by adding 4 points to each test score. Compute the sample mean based on the adjusted scores.
(e) Compare the unadjusted test score mean with the curved test score mean. What effect did adding 4 to each score have on the mean?

Kimberly Waterbury
Kimberly Waterbury
Numerade Educator
02:22

Problem 51

Another measure of central tendency is the trimmed mean. It is computed by determining the mean of a data set after deleting the smallest and largest observed values. Compute the trimmed mean for the data in Problem $33 .$ Is the trimmed mean resistant? Explain.

Jon Southam
Jon Southam
Numerade Educator
01:35

Problem 52

The midrange is also a measure of central tendency. It is computed by adding the smallest and largest observed values of a data set and dividing the result by $2 ;$ that is,
Midrange $=\frac{\text { largest data value }+\text { smallest data value }}{2}$
Compute the midrange for the data in Problem $33 .$ Is the midrange resistant? Explain.

Manisha Sarker
Manisha Sarker
Numerade Educator