15.8: Chapter 8
- Page ID
- 129937
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Your Turn
Major | Frequency |
---|---|
Biology | 6 |
Education | 1 |
Political Science | 3 |
Sociology | 2 |
Undecided | 4 |
Number of people in the residence | Frequency |
---|---|
1 | 12 |
2 | 13 |
3 | 8 |
4 | 6 |
5 | 1 |
Age range | Frequency |
---|---|
25-29 | 2 |
30-34 | 6 |
35-39 | 2 |
40-44 | 4 |
45-49 | 1 |
50-54 | 2 |
55-59 | 2 |
60-64 | 0 |
65-70 | 1 |
(Answers may vary depending on bin boundary decisions)
Major | Frequency | Proportion |
---|---|---|
Biology | 6 | 37.5% |
Education | 1 | 6.3% |
Political Science | 3 | 18.8% |
Sociology | 2 | 12.5% |
Undecided | 4 | 25% |
![A bar graph plots percentages of different majors. The horizontal axis represents majors. The vertical axis representing percent ranges from 0 percent to 40 percent, in increments of 5 percent. The graph infers the following data. Biology: 37.5 percent. Education: 6.3 percent. Political Science: 18.8 percent. Sociology: 12.5 percent. Undecided: 25 percent.](https://math.libretexts.org/@api/deki/files/110066/CS_Figure_08_02_010.jpg?revision=1)
![A pie chart represents the majors of students in the class. The pie chart infers the following data. Biology: 37.5 percent. Undecided: 25 percent. Political Science: 18.8 percent. Sociology: 12.5 percent. Education: 6.3 percent.](https://math.libretexts.org/@api/deki/files/110071/CS_Figure_08_02_014.jpg?revision=1)
4 | 7 |
5 | 9 7 4 |
6 | 9 8 7 |
7 | 8 7 5 2 2 1 0 |
8 | 9 6 5 4 4 1 |
9 | 7 7 6 3 3 1 |
10 | 7 6 3 1 |
![A histogram represents wins by MLB teams, 2019 season (good). The horizontal axis representing wins ranges from 40 to 110, in increments of 10. The vertical axis representing frequency ranges from 0 to 8, in increments of 2. The histogram infers the following data. 40 to 50: 1. 50 to 60: 3. 60 to 70: 3. 70 to 80: 7. 80 to 90: 6. 90 to 100: 6. 100 to 110: 4.](https://math.libretexts.org/@api/deki/files/110069/CS_Figure_08_02_019.jpg?revision=1)
Answers may vary based on bin choices. Here’s the result for bins of width 2,500:
![A histogram of in-state tuition costs at US institutions. The horizontal axis representing tuition ranges from 2500 to 75000, in increments of 2500. The vertical axis representing frequency ranges from 0 to 800, in increments of 200. The histogram infers the following data: 2500, 230. 5000, 680. 7500, 480. 10000, 390. 12500, 300. 15000, 390. 17500, 270. 20000, 175. 22500, 150. 25000, 100. 27500, 120. 30000, 150. 32500, 160. 35000, 120. 37500, 100. 40000, 90. 42500, 80. 45000, 70. 47500, 60. 50000, 50. 52500, 50. 55000, 70. 57500, 50. 60000, 10. 75000, 10. Note: all values are approximate.](https://math.libretexts.org/@api/deki/files/110070/CS_Figure_08_02_021.jpg?revision=1)
The data are strongly right-skewed.
![A bar graph represents world records for women’s swimming events, 100 meters. The horizontal axis represents the event. The vertical axis representing time in seconds ranges from 0 to 70, in increments of 10. The bar graph infers the following data. Freestyle: 52. Backstroke: 57. Breaststroke: 64. Butterfly: 56. Note: all values are approximate.](https://math.libretexts.org/@api/deki/files/110068/CS_Figure_08_02_023.jpg?revision=1)
Top ten teams by wins:
2
Median: 82.5
Mean: 80.967 (rounded to three decimal places)
/**/s = \sqrt {752.5} \approx 27.432/**/
The red (leftmost) distribution has mean 11, the blue (middle) has mean 13, and the yellow (rightmost) has mean 14.
65 and 75
26 and 54
110 and 290
47.5%
15.85%
81.5%
/**/−1.8/**/
/**/1.6/**/
/**/−0.6/**/
30
6
80th
9th
97th
3.9
6.3
2.9
Using NORM.INV: 1092.8
Using PERCENTILE: 1085
![A scatter plot shows five points. The horizontal axis representing receptions ranges from 105 to 130, in increments of 5. The vertical axis representing yards ranges from 1100 to 1600, in increments of 100. The points are at the following coordinates: (105, 1420), (107, 1200), (115, 1380), (115, 1410), and (127, 1540). Note: all values are approximate.](https://math.libretexts.org/@api/deki/files/110076/CS_Figure_08_08_107.jpg?revision=1)
![A scatter plot represents PTS versus W. The horizontal axis representing W ranges from 25 to 65, in increments of 5. The vertical axis representing PTS ranges from 50 to 130, in increments of 20. The points are arranged in increasing order. Some of the points are as follows: (32, 70), (35, 80), (40, 88), (45, 100), and (50, 108). Note: all values are approximate.](https://math.libretexts.org/@api/deki/files/110079/CS_Figure_08_08_109.jpg?revision=1)
- No curved pattern
- Strong negative relationship, /**/r \approx - 0.9/**/
- Curved pattern
- No curved pattern
- No apparent relationship, /**/r \approx 0/**/
- No curved pattern
- Weak positive relationship, /**/r \approx 0.6/**/
- /**/r = - 0.91/**/
- Not appropriate
- /**/r = - 0.01/**/
- /**/r = 0.62/**/
/**/y = 1.8x + 16.9/**/
/**/y = 0.174x + 14.5/**/
26.68
Predicted: 28; actual: 18. The Phillies were caught around 10 fewer times than expected.
Every 10 additional steal attempts will result in getting caught about 1.7 times on average.
Check Your Understanding
Genre | Frequency |
---|---|
Cooking | 1 |
Non-fiction | 3 |
Romance | 4 |
Thriller | 3 |
True Crime | 3 |
Young Adult | 6 |
Number of classes | Frequency |
---|---|
1 | 1 |
2 | 3 |
3 | 16 |
4 | 8 |
5 | 4 |
Range of Cell Phone Subscriptions Per Hundred People | Frequency |
---|---|
0.0 – 24.9 | 1 |
25.0 – 49.9 | 3 |
50.0 – 74.9 | 1 |
75.0 – 99.9 | 6 |
100.0 – 124.9 | 7 |
125.0 – 149.9 | 3 |
150.0 – 174.9 | 3 |
175.0 – 199.9 | 1 |
(Note: Answers may vary based on choices made about bins.)
![A bar graph titled, animals treated. The horizontal axis represents classification. The vertical axis representing frequency ranges from 0 to 14, in increments of 2. The bar graph infers the following data. Amphibian: 3. Bird: 5. Mammal: 12. Reptile: 4.](https://math.libretexts.org/@api/deki/files/110077/CS_Figure_08_02_033.jpg?revision=1)
![A pie chart titled, animals treated. The pie chart is divided into four unequal parts. The pie chart infers the following data. Amphibian: 3. Bird: 5. Mammal: 12. Reptile: 4.](https://math.libretexts.org/@api/deki/files/110078/CS_Figure_08_02_034.jpg?revision=1)
12 | 7 |
13 | 0 2 3 6 6 7 8 9 9 |
14 | 1 2 3 3 6 8 8 |
15 | 3 3 5 6 6 6 7 7 8 8 |
16 | 4 7 8 |
![A histogram titled, weekly help desk customers. The horizontal axis representing customers ranges from 125 to 170, in increments of 5. The vertical axis representing frequency ranges from 0 to 8, in increments of 2. The histogram infers the following data. 125 to 130: 1. 130 to 135: 3. 135 to 140: 6. 140 to 145: 4. 145 to 150: 3. 150 to 155: 2. 155 to 160: 8. 160 to 165: 1. 165 to 170: 2.](https://math.libretexts.org/@api/deki/files/110080/CS_Figure_08_02_035.jpg?revision=1)
![A bar graph titled, admission rate at different campuses in the University of California System. The horizontal axis represents campus. The vertical axis representing the admission rate ranges from 0 to 0.7, in increments of 0.1. The bar graph infers the following data. Berkeley: 0.14. Davis: 0.41. Irvine: 0.28. Los Angeles: 0.14. Merced: 0.66. Riverside: 0.5. San: Diego: 0.3. Santa Barbara: 0.32. Santa Cruz: 0.48. Note: all values are approximate.](https://math.libretexts.org/@api/deki/files/110081/CS_Figure_08_02_036.jpg?revision=1)
Mode: 112
Median: 113
Mean: 112.64
156
147
147.2
Mode: not useful; every value appears only once
Median: 0.322
Mean: 0.3612
Mode: not useful; every value appears only once
Median: $42,952
Mean: $42924.78
Median: $11,207
Mean: $15,476.79
23.2
1300 on the SAT
PERCENTILE gives 20.86; NORM.INV gives 21.11.
PERCENTILE gives 11.82; NORM.INV gives 11.68.
PERCENTRANK gives 92.6th; NORM.DIST gives 91.9th.
PERCENTRANK gives 77th; NORM.DIST gives 79.2nd.
![A scatter plot shows five points. The x-axis ranges from 5 to 30, in increments of 5. The y-axis ranges from 0 to 20, in increments of 5. The points are as follows: (8, 17), (11, 15), (20, 13), (22, 13), and (25, 10). Note: all values are approximate.](https://math.libretexts.org/@api/deki/files/110084/CS_Figure_08_08_133.jpg?revision=1)
- Yes
- No
- Weak positive relationship; /**/r \approx 0.5/**/
0.96
$7,475
Less than expected by $1,495.68
For every $1,000 increase in out-of-state tuition, we expect average monthly salary to increase by $161.