Understandable Statistics, Concepts and Methods

Charles Henry Brase, Corrinne Pellillo Brase

Chapter 2 Organizing Data - all with Video Answers

Educators

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Section 1

Frequency Distributions, Histograms, and Related Topics

01:47

Problem 1

What is the difference between a class boundary and a class limit?

Jerrah Biggerstaff

Numerade Educator

01:12

Problem 2

A data set has values ranging from a low of 10 to a high of $52 .$ What's wrong with using the class limits $10-19,20-29,30-39,40-49$ for a frequency table?

Jerrah Biggerstaff

Numerade Educator

01:25

Problem 3

A data set has values ranging from a low of 10 to a high of $50 .$ What's wrong with using the class limits $10-20,20-30,30-40,40-50$ for a frequency table?

Jerrah Biggerstaff

Numerade Educator

01:24

A data set has values ranging from a low of 10 to a high of $50 .$ The class width is to be $10 .$ What's wrong with using the class limits $10-20,21-31,32-42,43-53$ for a frequency table with a class width of $10 ?$

Jerrah Biggerstaff

Numerade Educator

01:18

Problem 5

Class Limits A data set with whole numbers has a low value of 20 and a high value of $82 .$ Find the class width and class limits for a frequency table with 7 classes.

Jerrah Biggerstaff

Numerade Educator

01:45

Problem 6

Class Limits A data set with whole numbers has a low value of 10 and a high value of $120 .$ Find the class width and class limits for a frequency table with 5 classes.

Jerrah Biggerstaff

Numerade Educator

01:05

Problem 7

You are manager of a specialty coffee shop and collect data throughout a full day regarding waiting time for customers from the time they enter the shop until the time they pick up their order.
(a) What type of distribution do you think would be most desirable for the waiting times: skewed right, skewed left, mound-shaped symmetric? Explain.
(b) What if the distribution for waiting times were bimodal? What might be some explanations?

Nick Johnson

Numerade Educator

01:33

Problem 8

A web site rated 100 colleges and ranked the colleges from 1 to 100, with a rank of 1 being the best. Each college was ranked, and there were no ties. If the ranks were displayed in a histogram, what would be the shape of the histogram: skewed, uniform, mound-shaped?

Jerrah Biggerstaff

Numerade Educator

04:57

Problem 9

Look at the histogram in Figure $2-10(a),$ which shows mileage, in miles per gallon (mpg), for a random selection of older passenger cars (Reference: Consumer Reports).
(a) Is the shape of the histogram essentially bimodal?
(b) Jose looked at the raw data and discovered that the 54 data values included both the city and the highway mileages for 27 cars. He used the city mileages for the 27 cars to make the histogram in Figure $2-10$ (b). Using this information and Figure $2-10,$ parts (a) and (b), construct a histogram for the highway mileages of the same cars. Use class boundaries 16.5,20.5,24.5 $28.5,32.5,36.5,$ and 40.5

Ahmad Reda

Numerade Educator

07:30

Problem 10

The following data represent annual salaries, in thousands of dollars, for employees of a small company. Notice that the data have been sorted in increasing order.
$$\begin{array}{ccccccccccccc}
54 & 55 & 55 & 57 & 57 & 59 & 60 & 65 & 65 & 65 & 66 & 68 & 68 \\
69 & 69 & 70 & 70 & 70 & 75 & 75 & 75 & 75 & 77 & 82 & 82 & 82 \\
88 & 89 & 89 & 91 & 91 & 97 & 98 & 98 & 98 & 280 & &
\end{array}$$
(a) Make a histogram using the class boundaries 53.5,99.5,145.5,191.5 237.5,283.5
(b) Look at the last data value. Does it appear to be an outlier? Could this be the owner's salary?
(c) Eliminate the high salary of 280 thousand dollars. Make a new histogram using the class boundaries $53.5,62.5,71.5,80.5,89.5,98.5 .$ Does this histogram reflect the salary distribution of most of the employees better than the histogram in part (a)?

Ahmad Reda

Numerade Educator

06:39

Problem 11

Histograms of random sample data are often used as an indication of the shape of the underlying population distribution. The histograms on the next page are based on random samples of size $30,50,$ and 100 from the same population.
(a) Using the midpoint labels of the three histograms, what would you say about the estimated range of the population data from smallest to largest? Does the bulk of the data seem to be between 8 and 12 in all three histograms?
(b) The population distribution from which the samples were drawn is symmetric and mound-shaped, with the top of the mound at $10,95 \%$ of the data between 8 and $12,$ and $99.7 \%$ of data between 7 and $13 .$ How well does each histogram reflect these characteristics?

Ahmad Reda

Numerade Educator

03:02

Problem 12

The following histograms are based on different random samples of size 100 drawn from the same population.
(a) Identify the midpoint of the class with the highest frequency in each of the three histograms.
(b) Using the class midpoints, what is the range of data shown in each histogram?
(c) Which of the histograms are more clearly skewed right?
(d) Based on your study of random sample in Chapter 1 , is it surprising to see the variations in the samples as displayed in the histograms? The original population from which the samples were drawn is skewed right with a high frequency near $4 .$ Do all three random samples seem to reflect these properties equally well?

Jerrah Biggerstaff

Numerade Educator

03:12

Problem 13

The ogives shown are based on U.S. Census data and show the average annual personal income per capita for each of the 50 states. The data are rounded to the nearest thousand dollars.
(a) How were the percentages shown in graph (ii) computed?
(b) How many states have average per capita income less than 37.5 thousand dollars?
(c) How many states have average per capita income between 42.5 and
52.5 thousand dollars?
(d) What percentage of the states have average per capita income more than
47.5 thousand dollars?

Bryan Meares

Numerade Educator

02:16

Problem 14

The following ogives come from different distributions of 50 whole numbers between 1 and 60. Labels on each point give the cumulative frequency and the cumulative percentage of data.
(a) In which distribution does the most data fall below 20.5?
(b) In which distribution does the most data fall below 40.5?
(c) In which distribution does the amount of data below 20.5 most closely match that above 30.5?
(d) Which distribution seems to be skewed right? Skewed left? Mound-shaped?

Bryan Meares

Numerade Educator

01:08

Problem 15

How long does it take to finish the 1161 -mile Iditarod Dog Sled Race from Anchorage to Nome, Alaska (see Viewpoint)? Finish times (to the nearest hour) for 57 dogsled teams are shown below.
$$\begin{aligned}
&\begin{array}{rrrrrrrrrrr}
261 & 271 & 236 & 244 & 279 & 296 & 284 & 299 & 288 & 288 & 247 & 256
\end{array}\\
&\begin{array}{rrrrrrrrrr}
338 & 360 & 341 & 333 & 261 & 266 & 287 & 296 & 313 & 311 & 307 & 307
\end{array}\\
&\begin{array}{cccccccccccc}
299 & 303 & 277 & 283 & 304 & 305 & 288 & 290 & 288 & 289 & 297 & 299
\end{array}\\
&\begin{array}{rrrrrrrrrr}
332 & 330 & 309 & 328 & 307 & 328 & 285 & 291 & 295 & 298 & 306 & 315
\end{array}\\
&\begin{array}{rrrrrrr}
310 & 318 & 318 & 320 & 333 & 321 & 323 & 324 & 327
\end{array}
\end{aligned}$$
Use five classes.

Nick Johnson

Numerade Educator

04:20

Problem 16

The following data represent glucose blood levels $(\mathrm{mg} / 100 \mathrm{ml})$ after a 12 -hour fast for a random sample of 70 women (Reference:
American Journal of Clinical Nutrition, Vol. 19, pp. 345-351). Note: These data are also available for download at the Companion Sites for this text.
$$\begin{aligned}
&\\
&\begin{array}{cccccccc}
45 & 66 & 83 & 71 & 76 & 64 & 59 & 59 \\
76 & 82 & 80 & 81 & 85 & 77 & 82 & 90 \\
87 & 72 & 79 & 69 & 83 & 71 & 87 & 69 \\
81 & 76 & 96 & 83 & 67 & 94 & 101 & 94 \\
89 & 94 & 73 & 99 & 93 & 85 & 83 & 80 \\
78 & 80 & 85 & 83 & 84 & 74 & 81 & 70 \\
65 & 89 & 70 & 80 & 84 & 77 & 65 & 46 \\
80 & 70 & 75 & 45 & 101 & 71 & 109 & 73 \\
73 & 80 & 72 & 81 & 63 & 74
\end{array}
\end{aligned}$$
Use six classes.

Sheryl Ezze

Numerade Educator

04:42

Problem 17

Certain kinds of tumors tend to recur. The following data represent the lengths of time, in months, for a tumor to recur after chemotherapy (Reference: D. P. Byar, Journal of Urology, Vol. 10, pp. $556-561$ ). Note: These data are also available for download at the Companion Sites for this text.
$$\begin{array}{ccccccccc}
19 & 18 & 17 & 1 & 21 & 22 & 54 & 46 & 25 & 49 \\
50 & 1 & 59 & 39 & 43 & 39 & 5 & 9 & 38 & 18 \\
14 & 45 & 54 & 59 & 46 & 50 & 29 & 12 & 19 & 36 \\
38 & 40 & 43 & 41 & 10 & 50 & 41 & 25 & 19 & 39 \\
27 & 20 & & & & &
\end{array}$$
Use five classes.

Neel Faucher

Numerade Educator

12:59

Problem 18

The Wind Mountain excavation site in New Mexico is an important archaeological location of the ancient Native American Anasazi culture. The following data represent depths (in cm) below surface grade at which significant artifacts were discovered at this site (Reference: A. I. Woosley and A. J. McIntyre, Mimbres Mogollon Archaeology, University of New Mexico Press). Note: These data are also available for download at the Companion Sites for this text.
$$\begin{array}{cccccccccc}
85 & 45 & 75 & 60 & 90 & 90 & 115 & 30 & 55 & 58 \\
78 & 120 & 80 & 65 & 65 & 140 & 65 & 50 & 30 & 125 \\
75 & 137 & 80 & 120 & 15 & 45 & 70 & 65 & 50 & 45 \\
95 & 70 & 70 & 28 & 40 & 125 & 105 & 75 & 80 & 70 \\
90 & 68 & 73 & 75 & 55 & 70 & 95 & 65 & 200 & 75 \\
15 & 90 & 46 & 33 & 100 & 65 & 60 & 55 & 85 & 50 \\
10 & 68 & 99 & 145 & 45 & 75 & 45 & 95 & 85 & 65 \\
65 & 52 & 82 & &
\end{array}$$
Use seven classes.

Carolyn Behr-Jerome

Numerade Educator

23:50

Problem 19

Education: College Enrollment What percent of undergraduate enrollment in coed colleges and universities in the United States is male? A random sample of 50 such institutions give the following data (Source USA Today College Guide).
Percent Males Enrolled in Coed Universities and Colleges
$$\begin{array}{rrrrrrrr}
31 & 39 & 53 & 47 & 40 & 49 & 53 & 47 \\
45 & 26 & 39 & 79 & 45 & 50 & 36 & 49 \\
45 & 49 & 43 & 48 & 54 & 50 & 43 & 42 \\
42 & 35 & 49 & 45 & 42 & 58 & 42 & 55 \\
45 & 71 & 50 & 57 & 49 & 50 & 45 & 46 \\
53 & 48 & 53 & 37 & 56 & 63 & 41 & 41 \\
51 & 48 & & & & & &
\end{array}$$
Use five classes.

Evelyn Cunningham

Numerade Educator

03:22

Problem 20

Advertising: Readability “Readability Levels of Magazine Ads," by F. K. Shuptrine and D. D. McVicker, is an article in the Journal of Advertising Research. (For more information, find the web site for DASL, the Carnegie Mellon University Data and Story Library. Look in Data Subjects under Consumer and then Magazine Ads Readability file.) The following is a list of the number of three-syllable (or longer) words in advertising copy of randomly selected magazine advertisements.
$$\begin{array}{ccccccccccc}
34 & 21 & 37 & 31 & 10 & 24 & 39 & 10 & 17 & 18 & 32 \\
17 & 3 & 10 & 6 & 5 & 6 & 6 & 13 & 22 & 25 & 3 \\
5 & 2 & 9 & 3 & 0 & 4 & 29 & 26 & 5 & 5 & 24 \\
15 & 3 & 8 & 16 & 9 & 10 & 3 & 12 & 10 & 10 & 10 \\
11 & 12 & 13 & 1 & 9 & 43 & 13 & 14 & 32 & 24 & 15
\end{array}$$
Use eight classes.

Kathleen Snyder

Numerade Educator

03:03

Problem 21

Decimal Data The following data represent tonnes of wheat harvested each year $(1894-1925)$ from Plot 19 at the Rothamsted Agricultural Experiment Stations, England.
$\begin{array}{ccccccccc}2.71 & 1.62 & 2.60 & 1.64 & 2.20 & 2.02 & 1.67 & 1.99 & 2.34 & 1.26 & 1.31\end{array}$
$\begin{array}{cccccccccc}1.80 & 2.82 & 2.15 & 2.07 & 1.62 & 1.47 & 2.19 & 0.59 & 1.48 & 0.77 & 2.04\end{array}$
$\begin{array}{cccccccccc}1.32 & 0.89 & 1.35 & 0.95 & 0.94 & 1.39 & 1.19 & 1.18 & 0.46 & 0.70\end{array}$
(a) Multiply each data value by 100 to "clear" the decimals.
(b) Use the standard procedures of this section to make a frequency table and histogram with your whole-number data. Use six classes.
(c) Divide class limits, class boundaries, and class midpoints by 100 to get back to your original data values.

Kathleen Snyder

Numerade Educator

03:21

Problem 22

Batting Averages The following data represent baseball batting averages for a random sample of National League players near the end of the baseball season. The data are from the baseball statistics section of the Denver Post
$$\begin{aligned}
&\begin{array}{cccccccc}
0.194 & 0.258 & 0.190 & 0.291 & 0.158 & 0.295 & 0.261 & 0.250 & 0.181
\end{array}\\
&\begin{array}{cccccccc}
0.125 & 0.107 & 0.260 & 0.309 & 0.309 & 0.276 & 0.287 & 0.317 & 0.252
\end{array}\\
&\begin{array}{ccccccc}
0.215 & 0.250 & 0.246 & 0.260 & 0.265 & 0.182 & 0.113 & 0.200
\end{array}
\end{aligned}$$
(a) Multiply each data value by 1000 to "clear" the decimals.
(b) Use the standard procedures of this section to make a frequency table and histogram with your whole-number data. Use five classes.
(c) Divide class limits, class boundaries, and class midpoints by 1000 to get back to your original data.

Kathleen Snyder

Numerade Educator

01:59

Problem 23

Another display technique that is somewhat similar to a histogram is a dotplot. In a dotplot, the data values are displayed along the horizontal axis. A dot is then plotted over each data value in the data set.
(a) From the dotplot, how many states have 600 or fewer licensed drivers per 1000 residents?
(b) About what percentage of the states (out of 51 ) seem to have close to 800 licensed drivers per 1000 residents?
(c) Consider the intervals 550 to 650,650 to $750,$ and 750 to 850 licensed drivers per 1000 residents. In which interval do most of the states fall?

Jerrah Biggerstaff

Numerade Educator

02:26

Problem 24

Make a dotplot for the data in Problem 15 regarding the finish time (number of hours) for the Iditarod Dog Sled Race. Compare the dotplot to the histogram of Problem $15 .$

Laurie Huffman

Numerade Educator

03:00

Problem 25

Make a dotplot for the data in Problem 17 regarding the recurrence of tumors after chemotherapy. Compare the dotplot to the histogram of Problem $17 .$

Jerrah Biggerstaff

Numerade Educator

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25