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Excursions in Modern Mathematics

Peter Tannenbaum

Chapter 15

Graphs, Charts, and Numbers - all with Video Answers

Educators

Chapter Questions

02:11

Problem 1

Refer to the data set shown in Table $15-12 .$ The table shows the scores on a Chem 103 test consisting of 10 questions worth 10 points each.
$$
\begin{array}{c|c|c|c}
\begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \text { Score } & \begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \text { Score } \\
\hline 1362 & 50 & 4315 & 70 \\
\hline 1486 & 70 & 4719 & 70 \\
\hline 1721 & 80 & 4951 & 60 \\
\hline 1932 & 60 & 5321 & 60 \\
\hline 2489 & 70 & 5872 & 100 \\
\hline 2766 & 10 & 6433 & 50 \\
\hline 2877 & 80 & 6921 & 50 \\
\hline 2964 & 60 & 8317 & 70 \\
\hline 3217 & 70 & 8854 & 100 \\
\hline 3588 & 80 & 8964 & 80 \\
\hline 3780 & 80 & 9158 & 60 \\
\hline 3921 & 60 & 9347 & 60 \\
\hline 4107 & 40 & &
\end{array}
$$
(a) Make a frequency table for the Chem 103 test scores.
(b) Draw a bar graph for the data in Table $15-12$.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
02:11

Problem 2

Refer to the data set shown in Table $15-12 .$ The table shows the scores on a Chem 103 test consisting of 10 questions worth 10 points each.
$$
\begin{array}{c|c|c|c}
\begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \text { Score } & \begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \text { Score } \\
\hline 1362 & 50 & 4315 & 70 \\
\hline 1486 & 70 & 4719 & 70 \\
\hline 1721 & 80 & 4951 & 60 \\
\hline 1932 & 60 & 5321 & 60 \\
\hline 2489 & 70 & 5872 & 100 \\
\hline 2766 & 10 & 6433 & 50 \\
\hline 2877 & 80 & 6921 & 50 \\
\hline 2964 & 60 & 8317 & 70 \\
\hline 3217 & 70 & 8854 & 100 \\
\hline 3588 & 80 & 8964 & 80 \\
\hline 3780 & 80 & 9158 & 60 \\
\hline 3921 & 60 & 9347 & 60 \\
\hline 4107 & 40 & &
\end{array}
$$
Draw a line graph for the data in Table $15-12$.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
01:10

Problem 3

Refer to the data set shown in Table $15-12 .$ The table shows the scores on a Chem 103 test consisting of 10 questions worth 10 points each.
$$
\begin{array}{c|c|c|c}
\begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \text { Score } & \begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \text { Score } \\
\hline 1362 & 50 & 4315 & 70 \\
\hline 1486 & 70 & 4719 & 70 \\
\hline 1721 & 80 & 4951 & 60 \\
\hline 1932 & 60 & 5321 & 60 \\
\hline 2489 & 70 & 5872 & 100 \\
\hline 2766 & 10 & 6433 & 50 \\
\hline 2877 & 80 & 6921 & 50 \\
\hline 2964 & 60 & 8317 & 70 \\
\hline 3217 & 70 & 8854 & 100 \\
\hline 3588 & 80 & 8964 & 80 \\
\hline 3780 & 80 & 9158 & 60 \\
\hline 3921 & 60 & 9347 & 60 \\
\hline 4107 & 40 & &
\end{array}
$$
Suppose that the grading scale for the test is $\mathrm{A}: 80-100 ; \mathrm{B}:$ $70-79 ; \mathrm{C}: 60-69 ; \mathrm{D}: 50-59 ;$ and $\mathrm{F}: 0-49$
(a) Make a frequency table for the distribution of the test grades.
(b) Draw a relative frequency bar graph for the test grades.

CG
Christie Gilbert
Numerade Educator
View

Problem 4

Refer to the data set shown in Table $15-12 .$ The table shows the scores on a Chem 103 test consisting of 10 questions worth 10 points each.
$$
\begin{array}{c|c|c|c}
\begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \text { Score } & \begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \text { Score } \\
\hline 1362 & 50 & 4315 & 70 \\
\hline 1486 & 70 & 4719 & 70 \\
\hline 1721 & 80 & 4951 & 60 \\
\hline 1932 & 60 & 5321 & 60 \\
\hline 2489 & 70 & 5872 & 100 \\
\hline 2766 & 10 & 6433 & 50 \\
\hline 2877 & 80 & 6921 & 50 \\
\hline 2964 & 60 & 8317 & 70 \\
\hline 3217 & 70 & 8854 & 100 \\
\hline 3588 & 80 & 8964 & 80 \\
\hline 3780 & 80 & 9158 & 60 \\
\hline 3921 & 60 & 9347 & 60 \\
\hline 4107 & 40 & &
\end{array}
$$
Suppose that the grading scale for the test is $\mathrm{A}: 80-100 ; \mathrm{B}:$ $70-79 ; \mathrm{C}: 60-69 ; \mathrm{D}: 50-59 ;$ and $\mathrm{F}: 0-49 .$
(a) What percentage of the students who took the test got a grade of D?
(b) In a pie chart showing the distribution of the test grades, what is the size of the central angle (in degrees) of the "wedge" representing the grade of D?
(c) Draw a pie chart showing the distribution of the test grades. Give the central angles for each wedge in the pie chart (round your answer to the nearest degree).

Lainey Roebuck
Lainey Roebuck
Numerade Educator
02:26

Problem 5

Refer to Table $15-13,$ which gives the home-to-school distance $d$ (rounded to the nearest half-mile) for each of the 27 kindergarten students at Cleansburg Elementary School.
$$
\begin{array}{c|c|c|c}
\begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \boldsymbol{d} & \begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \boldsymbol{d} \\
\hline 1362 & 1.5 & 3921 & 5.0 \\
\hline 1486 & 2.0 & 4355 & 1.0 \\
\hline 1587 & 1.0 & 4454 & 1.5 \\
\hline 1877 & 0.0 & 4561 & 1.5 \\
\hline 1932 & 1.5 & 5482 & 2.5 \\
\hline 1946 & 0.0 & 5533 & 1.5 \\
\hline 2103 & 2.5 & 5717 & 8.5 \\
\hline 2877 & 1.0 & 6307 & 1.5 \\
\hline 2964 & 0.5 & 6573 & 0.5 \\
\hline 3491 & 0.0 & 8436 & 3.0 \\
\hline 3588 & 0.5 & 8592 & 0.0 \\
\hline 3711 & 1.5 & 8964 & 2.0 \\
\hline 3780 & 2.0 & 9205 & 0.5 \\
\hline & & 9658 & 6.0 \\
\hline
\end{array}
$$
(a) Make a frequency table for the distances in Table $15-13 .$
(b) Draw a line graph for the data in Table $15-13$.

James York
James York
Numerade Educator
04:15

Problem 6

Refer to Table $15-13,$ which gives the home-to-school distance $d$ (rounded to the nearest half-mile) for each of the 27 kindergarten students at Cleansburg Elementary School.
$$
\begin{array}{c|c|c|c}
\begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \boldsymbol{d} & \begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \boldsymbol{d} \\
\hline 1362 & 1.5 & 3921 & 5.0 \\
\hline 1486 & 2.0 & 4355 & 1.0 \\
\hline 1587 & 1.0 & 4454 & 1.5 \\
\hline 1877 & 0.0 & 4561 & 1.5 \\
\hline 1932 & 1.5 & 5482 & 2.5 \\
\hline 1946 & 0.0 & 5533 & 1.5 \\
\hline 2103 & 2.5 & 5717 & 8.5 \\
\hline 2877 & 1.0 & 6307 & 1.5 \\
\hline 2964 & 0.5 & 6573 & 0.5 \\
\hline 3491 & 0.0 & 8436 & 3.0 \\
\hline 3588 & 0.5 & 8592 & 0.0 \\
\hline 3711 & 1.5 & 8964 & 2.0 \\
\hline 3780 & 2.0 & 9205 & 0.5 \\
\hline & & 9658 & 6.0 \\
\hline
\end{array}
$$
Draw a bar graph for the data in Table $15-13$.

Ashley Volpe
Ashley Volpe
Numerade Educator
02:17

Problem 7

Refer to Table $15-13,$ which gives the home-to-school distance $d$ (rounded to the nearest half-mile) for each of the 27 kindergarten students at Cleansburg Elementary School.
$$
\begin{array}{c|c|c|c}
\begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \boldsymbol{d} & \begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \boldsymbol{d} \\
\hline 1362 & 1.5 & 3921 & 5.0 \\
\hline 1486 & 2.0 & 4355 & 1.0 \\
\hline 1587 & 1.0 & 4454 & 1.5 \\
\hline 1877 & 0.0 & 4561 & 1.5 \\
\hline 1932 & 1.5 & 5482 & 2.5 \\
\hline 1946 & 0.0 & 5533 & 1.5 \\
\hline 2103 & 2.5 & 5717 & 8.5 \\
\hline 2877 & 1.0 & 6307 & 1.5 \\
\hline 2964 & 0.5 & 6573 & 0.5 \\
\hline 3491 & 0.0 & 8436 & 3.0 \\
\hline 3588 & 0.5 & 8592 & 0.0 \\
\hline 3711 & 1.5 & 8964 & 2.0 \\
\hline 3780 & 2.0 & 9205 & 0.5 \\
\hline & & 9658 & 6.0 \\
\hline
\end{array}
$$
Draw a bar graph for the home-to-school distances for the kindergarteners at Cleansburg Elementary School using the following class intervals:
Very close: Less than 1 mile
Close: 1 mile up to and including 1.5 miles
Nearby: 2 miles up to and including 2.5 miles
Not too far: 3 miles up to and including 4.5 miles
Far: 5 miles or more

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:26

Problem 8

Refer to Table $15-13,$ which gives the home-to-school distance $d$ (rounded to the nearest half-mile) for each of the 27 kindergarten students at Cleansburg Elementary School.
$$
\begin{array}{c|c|c|c}
\begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \boldsymbol{d} & \begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \boldsymbol{d} \\
\hline 1362 & 1.5 & 3921 & 5.0 \\
\hline 1486 & 2.0 & 4355 & 1.0 \\
\hline 1587 & 1.0 & 4454 & 1.5 \\
\hline 1877 & 0.0 & 4561 & 1.5 \\
\hline 1932 & 1.5 & 5482 & 2.5 \\
\hline 1946 & 0.0 & 5533 & 1.5 \\
\hline 2103 & 2.5 & 5717 & 8.5 \\
\hline 2877 & 1.0 & 6307 & 1.5 \\
\hline 2964 & 0.5 & 6573 & 0.5 \\
\hline 3491 & 0.0 & 8436 & 3.0 \\
\hline 3588 & 0.5 & 8592 & 0.0 \\
\hline 3711 & 1.5 & 8964 & 2.0 \\
\hline 3780 & 2.0 & 9205 & 0.5 \\
\hline & & 9658 & 6.0 \\
\hline
\end{array}
$$
Draw a bar graph for the home-to-school distances for the kindergarteners at Cleansburg Elementary School using the following class intervals:
Zone $A: 1.5$ miles or less
Zone $B$ : more than 1.5 miles up to and including 2.5 miles

Zone $C$ : more than 2.5 miles up to and including 3.5 miles
Zone $D$ : more than 3.5 miles

James York
James York
Numerade Educator
00:51

Problem 9

Refer to Table $15-13,$ which gives the home-to-school distance $d$ (rounded to the nearest half-mile) for each of the 27 kindergarten students at Cleansburg Elementary School.
$$
\begin{array}{c|c|c|c}
\begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \boldsymbol{d} & \begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \boldsymbol{d} \\
\hline 1362 & 1.5 & 3921 & 5.0 \\
\hline 1486 & 2.0 & 4355 & 1.0 \\
\hline 1587 & 1.0 & 4454 & 1.5 \\
\hline 1877 & 0.0 & 4561 & 1.5 \\
\hline 1932 & 1.5 & 5482 & 2.5 \\
\hline 1946 & 0.0 & 5533 & 1.5 \\
\hline 2103 & 2.5 & 5717 & 8.5 \\
\hline 2877 & 1.0 & 6307 & 1.5 \\
\hline 2964 & 0.5 & 6573 & 0.5 \\
\hline 3491 & 0.0 & 8436 & 3.0 \\
\hline 3588 & 0.5 & 8592 & 0.0 \\
\hline 3711 & 1.5 & 8964 & 2.0 \\
\hline 3780 & 2.0 & 9205 & 0.5 \\
\hline & & 9658 & 6.0 \\
\hline
\end{array}
$$
Using the class intervals given in Exercise $7,$ draw a pie chart for the home-to-school distances for the kindergarteners at Cleansburg Elementary School. Give the central angles for each wedge of the pie chart. Round your answer to the nearest degree.

Christopher Stanley
Christopher Stanley
Numerade Educator
00:51

Problem 10

Refer to Table $15-13,$ which gives the home-to-school distance $d$ (rounded to the nearest half-mile) for each of the 27 kindergarten students at Cleansburg Elementary School.
$$
\begin{array}{c|c|c|c}
\begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \boldsymbol{d} & \begin{array}{c}
\text { Student } \\
\text { ID }
\end{array} & \boldsymbol{d} \\
\hline 1362 & 1.5 & 3921 & 5.0 \\
\hline 1486 & 2.0 & 4355 & 1.0 \\
\hline 1587 & 1.0 & 4454 & 1.5 \\
\hline 1877 & 0.0 & 4561 & 1.5 \\
\hline 1932 & 1.5 & 5482 & 2.5 \\
\hline 1946 & 0.0 & 5533 & 1.5 \\
\hline 2103 & 2.5 & 5717 & 8.5 \\
\hline 2877 & 1.0 & 6307 & 1.5 \\
\hline 2964 & 0.5 & 6573 & 0.5 \\
\hline 3491 & 0.0 & 8436 & 3.0 \\
\hline 3588 & 0.5 & 8592 & 0.0 \\
\hline 3711 & 1.5 & 8964 & 2.0 \\
\hline 3780 & 2.0 & 9205 & 0.5 \\
\hline & & 9658 & 6.0 \\
\hline
\end{array}
$$
Using the class intervals given in Exercise $8,$ draw a pie chart for the home-to-school distances for the kindergarteners at Cleansburg Elementary School. Give the central angles for each wedge of the pie chart. Round your answer to the nearest degree.

Christopher Stanley
Christopher Stanley
Numerade Educator
00:56

Problem 11

Refer to the bar graph shown in Fig. $15-15$ describing the scores of a group of students on a 10 -point math quiz.
(a) How many students took the math quiz?
(b) What percentage of the students scored 2 points?
(c) If a grade of 6 or more was needed to pass the quiz, what percentage of the students passed? (Round your answer to the nearest percent.)

NW
Nida Wasiq
Numerade Educator
00:56

Problem 11

Refer to the bar graph shown in Fig. $15-15$ describing the scores of a group of students on a 10 -point math quiz.
(a) How many students took the math quiz?
(b) What percentage of the students scored 2 points?
(c) If a grade of 6 or more was needed to pass the quiz, what percentage of the students passed? (Round your answer to the nearest percent.)

NW
Nida Wasiq
Numerade Educator
01:11

Problem 12

Refer to the bar graph shown in Fig. $15-15$ describing the scores of a group of students on a 10 -point math quiz.
Draw a relative frequency bar graph showing the results of the quiz.

Lucas Finney
Lucas Finney
Numerade Educator
01:11

Problem 12

Refer to the bar graph shown in Fig. $15-15$ describing the scores of a group of students on a 10 -point math quiz.
Draw a relative frequency bar graph showing the results of the quiz.

Lucas Finney
Lucas Finney
Numerade Educator
00:54

Problem 13

Refer to the pie chart in Fig. $15-16 .$
(a) Is cause of death a quantitative or a qualitative variable?
(b) Use the data provided in the pie chart to estimate the number of 18 - to 22 -year-olds who died in the United States in 2005 due to an accident.

Robin Corrigan
Robin Corrigan
Numerade Educator
00:54

Problem 14

Refer to the pie chart in Fig. $15-16 .$
Use the data provided in the pie chart to estimate the number of 18 - to 22 -year-olds who died in the United States in 2005 for each category shown in the pie chart.

Robin Corrigan
Robin Corrigan
Numerade Educator
06:08

Problem 15

Table $15-14$ shows the class interval frequencies for the 2015 Critical Reading scores on the SAT. Draw a relative frequency bar graph for the data in Table $15-14$. (Round the relative frequencies to the nearest tenth of a percent.)
$$
\begin{array}{c|c}
\text { Score range } & \text { Number of test-takers } \\
\hline 700-800 & 75,659 \\
\hline 600-690 & 257,184 \\
\hline 500-590 & 495,917 \\
\hline 400-490 & 540,157 \\
\hline 300-390 & 264,155 \\
\hline 200-290 & 65,449 \\
\hline \text { Total } & N=1,698,521
\end{array}
$$

Jerelyn Nevil
Jerelyn Nevil
Numerade Educator
06:08

Problem 16

Table $15-15$ shows the class interval frequencies for the 2015 Writing scores on the SAT. Draw a pie chart for the data in Table $15-15 .$ Indicate the degree of the central angle for each wedge of the pie chart (rounded to the nearest degree).$$
\begin{array}{c|c}
\text { Score range } & \text { Number of test-takers } \\
\hline 700-800 & 70,216 \\
\hline 600-690 & 229,224 \\
\hline 500-590 & 445,181 \\
\hline 400-490 & 575,463 \\
\hline 300-390 & 311,883 \\
\hline 200-290 & 66,554 \\
\hline \text { Total } & N=1,698,521
\end{array}
$$

Jerelyn Nevil
Jerelyn Nevil
Numerade Educator
01:27

Problem 17

Table $15-16$ shows the percentage of U.S. working married couples in which the wife's income is higher than the husband's $(1999-2009) .$
(a) Draw a pictogram for the data in Table $15-16$. Assume you are trying to convince your audience that things are looking great for women in the workplace and that women's salaries are catching up to men's very quickly.
(b) Draw a different pictogram for the data in Table $15-16$, where you are trying to convince your audience that women's salaries are catching up with men's very slowly.
$$
\begin{array}{l|c|c|c|c|c|c}
\text { Year } & 1999 & 2000 & 2001 & 2002 & 2003 & 2004 \\
\hline \text { Percent } & 28.9 & 29.9 & 30.7 & 31.9 & 32.4 & 32.6 \\
\hline \text { Year } & 2005 & 2006 & 2007 & 2008 & 2009 & \\
\hline \text { Percent } & 33.0 & 33.4 & 33.5 & 34.5 & 37.7 &
\end{array}
$$

Jerelyn Nevil
Jerelyn Nevil
Numerade Educator
01:20

Problem 18

Table $15-17$ shows the percentage of U.S. workers who are members of unions $(2000-2011)$.
(a) Draw a pictogram for the data in Table 15-17. Assume you are trying to convince your audience that unions are holding their own and that the percentage of union members in the workforce is steady.
(b) Draw a different pictogram for the data in Table $15-17$ where you are trying to convince your audience that there is a steep decline in union membership in the U.S. workforce.
$$
\begin{array}{l|r|r|r|r|r|r}
\text { Year } & 2000 & 2001 & 2002 & 2003 & 2004 & 2005 \\
\hline \text { Percent } & 13.4 & 13.3 & 13.3 & 12.9 & 12.5 & 12.5 \\
\hline \text { Year } & 2006 & 2007 & 2008 & 2009 & 2010 & 2011 \\
\hline \text { Percent } & 12.0 & 12.1 & 12.4 & 12.3 & 11.9 & 11.8
\end{array}
$$

Alexander Cheng
Alexander Cheng
Numerade Educator
27:59

Problem 19

Refer to Table $15-18,$ which shows the birth weights (in ounces) of the 625 babies born in Cleansburg hospitals in 2016.
$$
\begin{array}{c|c|c|c|c|c}
\begin{array}{c}
\text { More } \\
\text { than }
\end{array} & \begin{array}{c}
\text { Less } \\
\text { than or } \\
\text { equal } \\
\text { to }
\end{array} & \begin{array}{c}
\text { Num- } \\
\text { ber of } \\
\text { babies }
\end{array} & \begin{array}{c}
\text { More } \\
\text { than }
\end{array} & \begin{array}{c}
\text { Less } \\
\text { than or } \\
\text { equal } \\
\text { to }
\end{array} & \begin{array}{c}
\text { Num- } \\
\text { ber of } \\
\text { babies }
\end{array} \\
\hline 48 & 60 & 15 & 108 & 120 & 184 \\
\hline 60 & 72 & 24 & 120 & 132 & 142 \\
\hline 72 & 84 & 41 & 132 & 144 & 26 \\
\hline 84 & 96 & 67 & 144 & 156 & 5 \\
\hline 96 & 108 & 119 & 156 & 168 & 2
\end{array}
$$
(a) Give the length of each class interval (in ounces).
(b) Suppose that a baby weighs exactly 5 pounds 4 ounces. To what class interval does she belong? Describe the endpoint convention.
(c) Draw the histogram describing the 2016 birth weights in Cleansburg using the class intervals given in Table $15-18 .$

Mohan Jain
Mohan Jain
Numerade Educator
View

Problem 20

Refer to Table $15-18,$ which shows the birth weights (in ounces) of the 625 babies born in Cleansburg hospitals in 2016.
$$
\begin{array}{c|c|c|c|c|c}
\begin{array}{c}
\text { More } \\
\text { than }
\end{array} & \begin{array}{c}
\text { Less } \\
\text { than or } \\
\text { equal } \\
\text { to }
\end{array} & \begin{array}{c}
\text { Num- } \\
\text { ber of } \\
\text { babies }
\end{array} & \begin{array}{c}
\text { More } \\
\text { than }
\end{array} & \begin{array}{c}
\text { Less } \\
\text { than or } \\
\text { equal } \\
\text { to }
\end{array} & \begin{array}{c}
\text { Num- } \\
\text { ber of } \\
\text { babies }
\end{array} \\
\hline 48 & 60 & 15 & 108 & 120 & 184 \\
\hline 60 & 72 & 24 & 120 & 132 & 142 \\
\hline 72 & 84 & 41 & 132 & 144 & 26 \\
\hline 84 & 96 & 67 & 144 & 156 & 5 \\
\hline 96 & 108 & 119 & 156 & 168 & 2
\end{array}
$$
(a) Write a new frequency table for the birth weights in Cleansburg using class intervals of length equal to 24 ounces. Use the same endpoint convention as the one used in Table $15-18$.
(b) Draw the histogram corresponding to the frequency table found in (a).

Shu Naito
Shu Naito
Numerade Educator
04:08

Problem 21

Refer to the two histograms shown in Fig. 15 17 summarizing the 2016 payrolls of the 30 teams in Major League Baseball. The two histograms are based on the same data set but use slightly different class intervals. (You can assume that no team had a payroll that was exactly equal to $a$ whole number of millions of dollars.)
(a) How many teams had a 2016 payroll of more than $\$ 100$ million?
(b) How many teams had a 2016 payroll of less than $\$ 110$ million?
(c) How many teams had a 2016 payroll between $\$ 100$ and $\$ 110$ million?

Jon Southam
Jon Southam
Numerade Educator
04:08

Problem 22

Refer to the two histograms shown in Fig. 15 17 summarizing the 2016 payrolls of the 30 teams in Major League Baseball. The two histograms are based on the same data set but use slightly different class intervals. (You can assume that no team had a payroll that was exactly equal to $a$ whole number of millions of dollars.)
(a) How many teams had a 2016 payroll of less than $\$ 150$ million?
(b) How many teams had a 2016 payroll of more than $\$ 140$ million?
(c) How many teams had a 2016 payroll between $\$ 140$ and $\$ 150$ million?

Jon Southam
Jon Southam
Numerade Educator
04:08

Problem 23

Refer to the two histograms shown in Fig. 15 17 summarizing the 2016 payrolls of the 30 teams in Major League Baseball. The two histograms are based on the same data set but use slightly different class intervals. (You can assume that no team had a payroll that was exactly equal to $a$ whole number of millions of dollars.)
Consider the data set \{3,-5,7,4,8,2,8,-3,-6\} .
(a) Find the average $A$ of the data set.
(b) Find the median $M$ of the data set.
(c) Consider the data set \{3,-5,7,4,8,2,8,-3,-6,2\} obtained by adding one more data point to the original data set. Find the average and median of this data set.

Jon Southam
Jon Southam
Numerade Educator
04:08

Problem 24

Refer to the two histograms shown in Fig. 15 17 summarizing the 2016 payrolls of the 30 teams in Major League Baseball. The two histograms are based on the same data set but use slightly different class intervals. (You can assume that no team had a payroll that was exactly equal to $a$ whole number of millions of dollars.)
Consider the data set \{-4,6,8,-5.2,10.4,10,12.6,-13\}
(a) Find the average $A$ of the data set.
(b) Find the median $M$ of the data set.
(c) Consider the data set \{-4,6,8,-5.2,10.4,10,12.6\} having one less data point than the original set. Find the average and the median of this data set.

Jon Southam
Jon Southam
Numerade Educator
04:08

Problem 25

Refer to the two histograms shown in Fig. 15 17 summarizing the 2016 payrolls of the 30 teams in Major League Baseball. The two histograms are based on the same data set but use slightly different class intervals. (You can assume that no team had a payroll that was exactly equal to $a$ whole number of millions of dollars.)
Find the average $A$ and the median $M$ of each data set.
(a) \{0,1,2,3,4,5,6,7,8,9\}
(b) \{1,2,3,4,5,6,7,8,9\}
(c) \{1,2,3,4,5,6,7,8,9,10\}

Jon Southam
Jon Southam
Numerade Educator
04:08

Problem 26

Refer to the two histograms shown in Fig. 15 17 summarizing the 2016 payrolls of the 30 teams in Major League Baseball. The two histograms are based on the same data set but use slightly different class intervals. (You can assume that no team had a payroll that was exactly equal to $a$ whole number of millions of dollars.)
Find the average $A$ and the median $M$ of each data set.
(a) \{1,2,1,2,1,2,1,2,1,2\}
(b) \{1,2,3,4,1,2,3,4,1,2,3,4,1,2,3,4\}
(c) \{1,2,3,4,5,5,4,3,2,1\}

Jon Southam
Jon Southam
Numerade Educator
04:08

Problem 27

Refer to the two histograms shown in Fig. 15 17 summarizing the 2016 payrolls of the 30 teams in Major League Baseball. The two histograms are based on the same data set but use slightly different class intervals. (You can assume that no team had a payroll that was exactly equal to $a$ whole number of millions of dollars.)
Find the average $A$ and the median $M$ of each data set.
(a) \{5,10,15,20,25,60\}
(b) \{105,110,115,120,125,160\}

Jon Southam
Jon Southam
Numerade Educator
04:08

Problem 28

Refer to the two histograms shown in Fig. 15 17 summarizing the 2016 payrolls of the 30 teams in Major League Baseball. The two histograms are based on the same data set but use slightly different class intervals. (You can assume that no team had a payroll that was exactly equal to $a$ whole number of millions of dollars.)
Find the average $A$ and the median $M$ of each data set.
(a) \{5,10,15,20,25,30,35,40,45,50\}
(b) \{55,60,65,70,75,80,85,90,95,100\}

Jon Southam
Jon Southam
Numerade Educator
View

Problem 29

Table $15-19$ shows the results of a 5 -point musical aptitude test given to a group of first-grade students.
(a) Find the average aptitude score.
(b) Find the median aptitude score.

Tanvi Garg
Tanvi Garg
Numerade Educator
09:11

Problem 30

Table $15-20$ shows the ages of the firefighters in the Cleansburg Fire Department.
$$
\begin{array}{l|c|c|c|c|c}
\text { Age } & 25 & 27 & 28 & 29 & 30 \\
\hline \text { Frequency } & 2 & 7 & 6 & 9 & 15 \\
\hline \text { Age } & 31 & 32 & 33 & 37 & 39 \\
\hline \text { Frequency } & 12 & 9 & 9 & 6 & 4 \\
\hline
\end{array}
$$
(a) Find the average age of the Cleansburg firefighters rounded to two decimal places.
(b) Find the median age of the Cleansburg firefighters.

Samuel Goyette
Samuel Goyette
Numerade Educator
01:24

Problem 31

Table $15-21$ shows the relative frequencies of the scores of a group of students on a philosophy quiz.
$$
\begin{array}{l|c|c|c|c|c}
\text { Score } & 4 & 5 & 6 & 7 & 8 \\
\hline \begin{array}{l}
\text { Relative } \\
\text { frequency }
\end{array} & 7 \% & 11 \% & 19 \% & 24 \% & 39 \%
\end{array}
$$
(a) Find the average quiz score.
(b) Find the median quiz score.

Amrita Bhasin
Amrita Bhasin
Numerade Educator
05:16

Problem 32

Table $15-22$ shows the relative frequencies of the scores of a group of students on a 10 -point math quiz.
$$
\begin{array}{l|c|c|c|c|c|c|c}
\text { Score } & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline \begin{array}{l}
\text { Relative } \\
\text { frequency }
\end{array} & 8 \% & 12 \% & 16 \% & 20 \% & 18 \% & 14 \% & 12 \%
\end{array}
$$
(a) Find the average quiz score rounded to two decimal places.
(b) Find the median quiz score.

Trinity Steen
Trinity Steen
Numerade Educator
02:00

Problem 33

Consider the data set \{-5,7,4,8,2,8,-3,-6\}
(a) Find the first quartile $Q_{1}$ of the data set.
(b) Find the third quartile $Q_{3}$ of the data set.
(c) Consider the data set \{-5,7,4,8,2,8,-3,-6,2\} obtained by adding one more data point to the original data set. Find the first and third quartiles of this data set.

Kerry Thornton-Genova
Kerry Thornton-Genova
Numerade Educator
02:00

Problem 34

Consider the data set $\{-4,6,8,-5.2,10.4,10,12.6,-13\} .$
(a) Find the first quartile $Q_{1}$ of the data set.
(b) Find the third quartile $Q_{3}$ of the data set.
(c) Consider the data set \{-4,6,8,-5.2,10.4,10,12.6\} obtained by deleting one data point from the original data set. Find the first and third quartiles of this data set.

Kerry Thornton-Genova
Kerry Thornton-Genova
Numerade Educator
02:08

Problem 35

For each data set, find the 75 th and the 90 th percentiles.
(a) $\{1,2,3,4, \ldots, 98,99,100\}$
(b) $\{0,1,2,3,4, \ldots, 98,99,100\}$
(c) $\{1,2,3,4, \ldots, 98,99\}$
(d) $\{1,2,3,4, \ldots, 98\}$

AG
Ankit Gupta
Numerade Educator
02:08

Problem 36

For each data set, find the 10 th and the 25 th percentiles.
(a) $\{1,2,3, \ldots, 49,50,50,49, \ldots, 3,2,1\}$
(b) $\{1,2,3, \ldots, 49,50,49, \ldots, 3,2,1\}$
(c) $\{1,2,3, \ldots, 49,49, \ldots, 3,2,1\}$

AG
Ankit Gupta
Numerade Educator
08:47

Problem 37

This exercise refers to the age distribution in the Cleansburg Fire Department shown in Table $15-20$ (Exercise 30 ).
(a) Find the first quartile of the data set.
(b) Find the third quartile of the data set.
(c) Find the 90 th percentile of the data set.

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
04:47

Problem 38

This exercise refers to the math quiz scores shown in Table $15-22$ (Exercise 32).
(a) Find the first quartile of the data set.
(b) Find the third quartile of the data set.
(c) Find the 70 th percentile of the data set.

Luca Alexander
Luca Alexander
Numerade Educator
07:51

Problem 39

Refer to SAT test scores for $2014 .$ A total of $N=1,672,395$ college-bound students took the SAT in 2014 Assume that the test scores are sorted from lowest to highest and that the sorted data set is $\left\{d_{1}, d_{2}, \ldots, d_{1,672,395}\right\}$.
(a) Determine the position of the median $M$.
(b) Determine the position of the first quartile $Q_{1}$.
(c) Determine the position of the 80 th percentile.

Vaidik Stats
Vaidik Stats
Numerade Educator
07:51

Problem 40

Refer to SAT test scores for $2014 .$ A total of $N=1,672,395$ college-bound students took the SAT in 2014 Assume that the test scores are sorted from lowest to highest and that the sorted data set is $\left\{d_{1}, d_{2}, \ldots, d_{1,672,395}\right\}$.
(a) Determine the position of the third quartile $Q_{3}$.
(b) Determine the position of the 60 th percentile.

Vaidik Stats
Vaidik Stats
Numerade Educator
02:31

Problem 41

Consider the data set \{-5,7,4,8,2,8,-3,-6\}
(a) Find the five-number summary of the data set. (Hint: see Exercise 33 ).
(b) Draw a box plot for the data set.

Jessica Waggener
Jessica Waggener
Numerade Educator
02:31

Problem 42

Consider the data set $\{-4,6,8,-5.2,10.4,10,12.6,-13\} .$
(a) Find the five-number summary of the data set. (Hint: see Exercise 34 ).
(b) Draw a box plot for the data set.

Jessica Waggener
Jessica Waggener
Numerade Educator
01:12

Problem 43

This exercise refers to the distribution of ages in the Cleansburg Fire Department shown in Table $15-20$ (see Exercises 30 and 37 ).
(a) Find the five-number summary of the data set.
(b) Draw a box plot for the data set.

Kerry Thornton-Genova
Kerry Thornton-Genova
Numerade Educator
02:31

Problem 44

This exercise refers to the distribution of math quiz scores shown in Table $15-22$ (see Exercises 32 and 38 ).
(a) Find the five-number summary of the data set.
(b) Draw a box plot for the data set.

Jessica Waggener
Jessica Waggener
Numerade Educator
03:01

Problem 45

Refer to the two box plots in Fig. $15-18$ showing the starting salaries of Tasmania State University first-year graduates in agriculture and engineering. (These are the two box plots discussed in Example $15.15 .)$
(a) What is the median salary for agriculture majors?
(b) What is the median salary for engineering majors?
(c) Explain how we can tell that the median salary for engineering majors is the same as the third quartile salary for agriculture majors.

Nick Johnson
Nick Johnson
Numerade Educator
03:43

Problem 46

Refer to the two box plots in Fig. $15-18$ showing the starting salaries of Tasmania State University first-year graduates in agriculture and engineering. (These are the two box plots discussed in Example $15.15 .)$
(a) Fill in the blank: Of the 612 engineering graduates, at most $\longrightarrow$ had a starting salary greater than $\$ 45,000$.
(b) Fill in the blank: If there were 240 agriculture graduates with starting salaries of $\$ 35,000$ or less, the total number of agriculture graduates is approximately

Julie Silva
Julie Silva
Numerade Educator
02:18

Problem 47

For the data set $\{-5,7,4,8,2,8,-3,-6\},$ find
(a) the range.
(b) the interquartile range (see Exercise 33 ).

Blank Blank
Blank Blank
Numerade Educator
02:18

Problem 48

For the data set $\{-5,7,4,8,2,8,-3,-6\},$ find
(a) the range.
(b) the interquartile range (see Exercise 33 ).

Blank Blank
Blank Blank
Numerade Educator
04:53

Problem 49

A realty company has sold $N=341$ homes in the last year. The five-number summary for the sale prices is $\operatorname{Min}=\$ 97,000, \quad Q_{1}=\$ 115,000, \quad M=\$ 143,000, \quad Q_{3}=$
$\$ 156,000,$ and $\operatorname{Max}=\$ 249,000 .$
(a) Find the interquartile range of the home sale prices.
(b) How many homes sold for a price between $\$ 115,000$ and $\$ 156,000$ (inclusive)? (Note: If you don't believe that you have enough information to give an exact answer, you should give the answer in the form of "at

Yifan Xu
Yifan Xu
Numerade Educator
04:53

Problem 49

A realty company has sold $N=341$ homes in the last year. The five-number summary for the sale prices is $\operatorname{Min}=\$ 97,000, \quad Q_{1}=\$ 115,000, \quad M=\$ 143,000, \quad Q_{3}=$
$\$ 156,000,$ and $\operatorname{Max}=\$ 249,000$
(a) Find the interquartile range of the home sale prices.
(b) How many homes sold for a price between $\$ 115,000$ and $\$ 156,000$ (inclusive)? (Note: If you don't believe that you have enough information to give an exact answer, you should give the answer in the form of "at least $-$ or "at most $-$ ")

Yifan Xu
Yifan Xu
Numerade Educator
05:43

Problem 50

This exercise refers to the starting salaries of Tasmania State University first-year graduates in agriculture and engineering discussed in Exercises 45 and $46 .$
(a) Estimate the range for the starting salaries of agriculture majors.
(b) Estimate the interquartile range for the starting salaries of engineering majors.

Robin Corrigan
Robin Corrigan
Numerade Educator
05:43

Problem 50

This exercise refers to the starting salaries of Tasmania State University first-year graduates in agriculture and engineering discussed in Exercises 45 and $46 .$
(a) Estimate the range for the starting salaries of agriculture majors.
(b) Estimate the interquartile range for the starting salaries of engineering majors.

Robin Corrigan
Robin Corrigan
Numerade Educator
01:47

Problem 51

You should use the following definition of an outlier: An outlier is any data value that is above the third quartile by more than 1.5 times the IQR [Outlier $\left.>Q_{3}+1.5(I Q R)\right]$ or below the first quartile by more than 1.5 times the IQR [Outlier $\left.<Q_{1}-1.5(I Q R)\right] .$ (Note: There is no one universally agreed upon definition of an outlier; this is but one of several definitions used by statisticians.)
Suppose that the preceding definition of outlier is applied to the Stat 101 data set discussed in Example 15.14 .
(a) Fill in the blank: Any score bigger than $\underline{\text { is an }}$ outlier.
(b) Fill in the blank: Any score smaller than $\underline{\text { is an }}$ outlier.
(c) Find the outliers (if there are any) in the Stat 101 data set.

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 52

You should use the following definition of an outlier: An outlier is any data value that is above the third quartile by more than 1.5 times the IQR [Outlier $\left.>Q_{3}+1.5(I Q R)\right]$ or below the first quartile by more than 1.5 times the IQR [Outlier $\left.<Q_{1}-1.5(I Q R)\right] .$ (Note: There is no one universally agreed upon definition of an outlier; this is but one of several definitions used by statisticians.)
Using the preceding definition, find the outliers (if there are any) in the City of Cleansburg Fire Department data set discussed in Exercises 30 and $37 .$ (Hint: Do Exercise 37 first.)

Jason Gerber
Jason Gerber
Numerade Educator
View

Problem 53

You should use the following definition of an outlier: An outlier is any data value that is above the third quartile by more than 1.5 times the IQR [Outlier $\left.>Q_{3}+1.5(I Q R)\right]$ or below the first quartile by more than 1.5 times the IQR [Outlier $\left.<Q_{1}-1.5(I Q R)\right] .$ (Note: There is no one universally agreed upon definition of an outlier; this is but one of several definitions used by statisticians.)
The distribution of the heights (in inches) of 18 -year-old
U.S. males has first quartile $Q_{1}=67$ in. and third quartile $Q_{3}=71$ in. Using the preceding definition, determine which heights correspond to outliers.

Jason Gerber
Jason Gerber
Numerade Educator
View

Problem 54

You should use the following definition of an outlier: An outlier is any data value that is above the third quartile by more than 1.5 times the IQR [Outlier $\left.>Q_{3}+1.5(I Q R)\right]$ or below the first quartile by more than 1.5 times the IQR [Outlier $\left.<Q_{1}-1.5(I Q R)\right] .$ (Note: There is no one universally agreed upon definition of an outlier; this is but one of several definitions used by statisticians.)
The distribution of the heights (in inches) of 18 -year-old
U.S. females has first quartile $Q_{1}=62.5$ in. and third quartile $Q_{3}=66$ in. Using the preceding definition, determine which heights correspond to outliers.

Jason Gerber
Jason Gerber
Numerade Educator
07:32

Problem 55

The purpose is to practice computing standard deviations the old fashioned way (by hand). Granted, computing standard deviations this way is not the way it is generally done in practice; a good calculator (or a computer package) will do it much faster and more accurately. The point is that computing a few standard deviations the old-fashioned way should help you understand the concept a little better. If you use a calculator or a computer to answer these exercises, you are defeating their purpose.
Find the standard deviation of each of the following data sets.
(a) \{5,5,5,5\}
(b) \{0,5,5,10\}
(c) \{0,10,10,20\}
(d) \{1,2,3,4,5\}

Matthias Wuest
Matthias Wuest
Numerade Educator
03:02

Problem 56

The purpose is to practice computing standard deviations the old fashioned way (by hand). Granted, computing standard deviations this way is not the way it is generally done in practice; a good calculator (or a computer package) will do it much faster and more accurately. The point is that computing a few standard deviations the old-fashioned way should help you understand the concept a little better. If you use a calculator or a computer to answer these exercises, you are defeating their purpose.
Find the standard deviation of each of the following data sets.
(a) \{3,3,3,3\}
(b) \{0,6,6,8\}
(c) \{-6,0,0,18\}
(d) \{6,7,8,9,10\}

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:34

Problem 57

Refer to the mode of a data set. The mode of a data set is the data point that occurs with the highest frequency. When there are several data points (or categories) tied for the most frequent, each of them is a mode, but if all data points have the same frequency, rather than say that every data point is a mode, it is customary to say that there is no mode.
(a) Find the mode of the data set given by Table $15-20$ (Exercise 30).
(b) Find the mode of the data set given by Fig. $15-15$ (Exercises 11 and 12 ).

Anas Venkitta
Anas Venkitta
Numerade Educator
01:34

Problem 58

Refer to the mode of a data set. The mode of a data set is the data point that occurs with the highest frequency. When there are several data points (or categories) tied for the most frequent, each of them is a mode, but if all data points have the same frequency, rather than say that every data point is a mode, it is customary to say that there is no mode.
(a) Find the mode category for the data set described by the pie chart in Fig. $15-19(\mathrm{a})$.
(b) Find the mode category for the data set shown in Fig. $15-19(\mathrm{~b})$. If there is no mode, your answer should indicate so.

Anas Venkitta
Anas Venkitta
Numerade Educator
01:32

Problem 59

Mike's average on the first five exams in Econ $1 \mathrm{~A}$ is 88 What must he earn on the next exam to raise his overall average to $90 ?$

Aman Gupta
Aman Gupta
Numerade Educator
01:54

Problem 60

Explain each of the following statements regarding the median score in one of the SAT sections:
(a) If the number of test-takers $N$ is odd, then the median score must end in 0 .
(b) If the number of test-takers $N$ is even, then the median score can end in 0 or 5 , but the chances that it will end in 5 are very low.

Gregory Higby
Gregory Higby
Numerade Educator
02:48

Problem 61

In $2006,$ the median SAT score was the average of $d_{732,872}$ and $d_{732,873},$ where $\left\{d_{1}, d_{2}, \ldots, d_{N}\right\}$ denotes the data set of all SAT scores ordered from lowest to highest. Determine the number of students $N$ who took the SAT in 2006 .

Heena Haldankar
Heena Haldankar
Numerade Educator
00:47

Problem 62

In $2004,$ the third quartile of the SAT scores was $d_{1,064,256}$ where $\left\{d_{1}, d_{2}, \ldots, d_{N}\right\}$ denotes the data set of all SAT scores ordered from lowest to highest. Determine the number of students $N$ who took the SAT in 2004 .

Joseph Liao
Joseph Liao
Numerade Educator
00:56

Problem 63

(a) Give an example of 10 numbers with an average less than the median.
(b) Give an example of 10 numbers with a median less than the average.
(c) Give an example of 10 numbers with an average less than the first quartile.
(d) Give an example of 10 numbers with an average more than the third quartile.

Zach Steedman
Zach Steedman
Numerade Educator
00:49

Problem 64

Suppose that the average of 10 numbers is 7.5 and that the smallest of them is $\operatorname{Min}=3$
(a) What is the smallest possible value of $\operatorname{Max} ?$
(b) What is the largest possible value of $\operatorname{Max} ?$

Linh Vu
Linh Vu
Numerade Educator
02:17

Problem 65

Figure $15-20$ shows two different histograms summarizing the 2008 payrolls of the 30 teams in Major League Baseball. Using the information shown in the figure, it can be determined that the median payroll in Major League Baseball in 2008 falls somewhere between $\$ 70$ million and $\$ 80$ million. Explain how.

Ryan Mcalister
Ryan Mcalister
Numerade Educator
01:00

Problem 66

What happens to the five-number summary of the Stat 101 data set (see Example 15.14) if
(a) two points are added to each score?
(b) $10 \%$ is added to each score?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
04:12

Problem 67

Let $A$ denote the average and $M$ the median of the data set $\left\{x_{1}, x_{2}, x_{3}, \ldots, x_{N}\right\}$
Let $c$ be any constant.
(a) Find the average of the data set $\left\{x_{1}+c, x_{2}+c, x_{3}+\right.$ $\left.c, \ldots, x_{N}+c\right\}$ expressed in terms of $A$ and $c$
(b) Find the median of the data set $\left\{x_{1}+c, x_{2}+c, x_{3}+\right.$ $\left.c, \ldots, x_{N}+c\right\}$ expressed in terms of $M$ and $c$

Sneha Ravi
Sneha Ravi
Numerade Educator
02:34

Problem 68

Refer to the mode of a data set. The mode of a data set is the data point that occurs with the highest frequency. When there are several data points (or categories) tied for the most frequent, each of them is a mode, but if all data points have the same frequency, rather than say that every data point is a mode, it is customary to say that there is no mode.
Explain why the data sets $\left\{x_{1}, x_{2}, x_{3}, \ldots, x_{N}\right\}$ and $\left\{x_{1}+c, x_{2}+c, x_{3}+c, \ldots, x_{N}+c\right\}$ have
(a) the same range.
(b) the same standard deviation.

Hossam Mohamed
Hossam Mohamed
Numerade Educator
13:06

Problem 69

Refer to histograms with unequal class intervals. When sketching such histograms, the columns must be drawn so that the frequencies or percentages are proportional to the area of the column. Figure $15-21$ illustrates this. If the column over class interval 1 represents $10 \%$ of the population, then the column over class interval 2 , also representing $10 \%$ of the population, must be one-third as high, because the class interval is three times as large (Fig. $15-21$ ).
If the height of the column over the class interval $20-30$ is one unit and the column represents $25 \%$ of the population, then
(a) how high should the column over the interval $30-35$ be if $50 \%$ of the population falls into this class interval?
(b) how high should the column over the interval $35-45$ be if $10 \%$ of the population falls into this class interval?
(c) how high should the column over the interval $45-60$ be if $15 \%$ of the population falls into this class interval?

Carolyn Behr-Jerome
Carolyn Behr-Jerome
Numerade Educator
05:03

Problem 70

Refer to histograms with unequal class intervals. When sketching such histograms, the columns must be drawn so that the frequencies or percentages are proportional to the area of the column. Figure $15-21$ illustrates this. If the column over class interval 1 represents $10 \%$ of the population, then the column over class interval 2 , also representing $10 \%$ of the population, must be one-third as high, because the class interval is three times as large (Fig. $15-21$ ).
Two hundred senior citizens are tested for fitness and rated on their times on a one-mile walk. These ratings and associated frequencies are given in Table 15-23. Draw a histogram for these data based on the categories defined by the ratings in the table.
$$
\begin{array}{l|c|c}
\text { Time } & \text { Rating } & \text { Frequency } \\
\hline 6^{+} \text {to } 10 \text { minutes } & \text { Fast } & 10 \\
\hline 10^{+} \text {to } 16 \text { minutes } & \text { Fit } & 90 \\
\hline 16^{+} \text {to } 24 \text { minutes } & \text { Average } & 80 \\
\hline 24^{+} \text {to } 40 \text { minutes } & \text { Slow } & 20
\end{array}
$$

Pammi Eswari
Pammi Eswari
Numerade Educator
00:39

Problem 71

A data set is called constant if every value in the data set is the same. Explain why any data set with standard deviation 0 must be a constant data set.

Ernest Castorena
Ernest Castorena
Numerade Educator
02:43

Problem 72

Show that the standard deviation of any set of numbers is always less than or equal to the range of the set of numbers.

Bailey Brooks
Bailey Brooks
Numerade Educator
01:37

Problem 73

Show that if $A$ is the mean and $M$ is the median of the data set $\{1,2,3, \ldots, N\},$ then for all values of $N, A=M$

Jerrah Biggerstaff
Jerrah Biggerstaff
Numerade Educator
01:22

Problem 74

Suppose that the standard deviation of the data set $\left\{x_{1}, x_{2}, x_{3}, \ldots, x_{N}\right\}$ is $\sigma .$ Explain why the standard deviation of the data set $\left\{a \cdot x_{1}, a \cdot x_{2}, a \cdot x_{3}, \ldots, a \cdot x_{N}\right\}$ (where $a$
is a positive number) is $a \cdot \sigma .$

Sanchit Jain
Sanchit Jain
Numerade Educator
04:24

Problem 75

The Russian mathematician $\mathrm{P}$.
L. Chebyshev (1821-1894) showed that for any data set and any constant $k$ greater than $1,$ at least $1-\left(1 / k^{2}\right)$ of the data must lie within $k$ standard deviations on either side of the mean $A$. For example, when $k=2$, this says that $1-\frac{1}{4}=\frac{3}{4}$ (i.e., $\left.75 \%\right)$ of the data must lie within two standard deviations of $A$ (i.e., somewhere between $A-2 \sigma$ and $A+2 \sigma$ ).
(a) Using Chebyshev's theorem, what percentage of a data set must lie within three standard deviations of the mean?
(b) How many standard deviations on each side of the mean must we take to be assured of including $99 \%$ of the data?
(c) Suppose that the average of a data set is $A$. Explain why there is no number $k$ of standard deviations for which we can be certain that $100 \%$ of the data lies within $k$ standard deviations on either side of the $\operatorname{mean} A$

Sonam Khatri
Sonam Khatri
Numerade Educator