10.9: Practice and Exploration
- Page ID
- 64733
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- The data given below are the result of rolling a six-sided die fifty times.
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6 |
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1 |
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1 |
6 |
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1 |
6 |
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5 |
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5 |
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2 |
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6 |
6 |
5 |
Construct a frequency distribution for the data including the relative frequencies and the percentages. Interpret your results. Are there faces of the die that turn up more than others? Does the die seem fair to you?
- The data given below are the result of rolling a six-sided die fifty times.
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5 |
5 |
6 |
2 |
2 |
1 |
2 |
4 |
5 |
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5 |
6 |
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5 |
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5 |
5 |
5 |
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2 |
3 |
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5 |
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5 |
5 |
6 |
4 |
5 |
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6 |
3 |
5 |
5 |
2 |
5 |
4 |
6 |
6 |
5 |
Construct a frequency distribution for the data including the relative frequencies and the percentages. Interpret your results. Are there faces of the die that turn up more than others? Does the die seem fair to you?
- The table below shows a frequency distribution based on 60 observations of the gender identities of students who took part in a survey about campus life at a large university. The table has been left incomplete with a question mark denoting the missing entries. Using the entries of the table that have been provided, fill in the missing entries.
|
Identity |
Relative Frequency |
Frequency |
Percent |
|
Agender |
2 |
? |
3.33% |
|
Cisgender |
35 |
0.5833 |
? |
|
Genderfluid |
7 |
? |
? |
|
Genderqueer |
? |
0.0833 |
? |
|
Intersex |
1 |
? |
? |
|
Nonconforming |
? |
? |
? |
|
Transgender |
5 |
? |
? |
|
Total |
60 |
? |
100 |
- In a study of student debt, the total amount borrowed for undergraduate education and race and ethnicity were studied for 2015–2016 graduates. Frequency distributions of debt by race for students earning a master's degree are given in the table below. This table only includes the percentages. What general trends and conclusions do you make from these frequency distributions?
|
Amount Borrowed |
|||||
|
Race or Ethnicity |
No Debt |
$1 to $24,999 |
$25,000 to $49,999 |
$50,000 to $74,999 |
$75,000 or more |
|
White |
43% |
22% |
19% |
10% |
7% |
|
African American |
19% |
18% |
26% |
22% |
16% |
|
Hispanic |
27% |
24% |
24% |
16% |
8% |
|
Asian |
50% |
11% |
14% |
12% |
13% |
- The table below shows a frequency distribution for the number of books checked out per week by 250 randomly sampled patrons who checked out at least one book at an inner-city library.
|
Number |
Frequency |
Relative Frequency |
Percent |
|
1 |
15 |
0.060 |
6.0% |
|
2 |
39 |
0.156 |
15.6% |
|
3 |
64 |
0.256 |
25.6% |
|
4 |
50 |
0.200 |
20.0% |
|
5 |
26 |
0.104 |
10.4% |
|
Over 5 |
56 |
0.224 |
22.4% |
|
Total |
60 |
1.000 |
100% |
What trends do you see in the table, and what types of conclusions can you make about the number of books checked out buy these individuals
?6. In a famous study of potential gender bias (Bickel et al. 1975), a cross-classified frequency distribution of data on gender and admission to the Berkeley graduate school for the fall 1973 semester are given below:
|
Gender |
Admit |
Deny |
Total |
|
Female |
1494 |
2827 |
4321 |
|
Male |
3738 |
4704 |
8442 |
|
Total |
5232 |
7531 |
12763 |
Compute the relative frequencies and percentages for this frequency distribution. What percentage of female applicants are admitted and how does this compare to the percentage of male applicants that are admitted? What conclusions can be drawn from this table?
- Continuing with the study on potential gender bias from the previous problem, the frequency table has been split up across the six largest majors that the female applicants applied to below:
|
Major |
Admit |
Deny |
Total |
|
A |
89 |
19 |
108 |
|
B |
17 |
8 |
25 |
|
C |
202 |
391 |
593 |
|
D |
131 |
244 |
375 |
|
E |
94 |
299 |
393 |
|
F |
24 |
317 |
341 |
|
Total |
557 |
1278 |
1835 |
and for the male applicants below:
|
Major |
Admit |
Deny | Total |
|
A |
512 |
313 | 825 |
|
B |
353 |
207 | 560 |
|
C |
120 |
205 | 325 |
|
D |
138 |
279 | 417 |
|
E |
53 |
138 | 191 |
|
F |
22 |
351 | 373 |
|
Total |
1198 |
1493 | 2691 |
Specific majors are not identified due to university policy (Freedman et al. 2007). Compute the relative frequencies and percentages for this frequency distributions. Looking at the trends in the frequency tables, does there seem to be gender bias in the admittance of students?
- The table below shows the frequency distributions for the number of students referred, the total enrollment, and race or ethnicity of students in the school-wide information system study for middle school students. This is additional data for the study considered earlier.
|
Race or Ethnicity |
Referred |
Enrollment |
|
Hispanic |
4,245 |
10,332 |
|
0.169 |
0.171 |
|
|
16.9% |
17.1% |
|
|
African American |
8,024 |
12,228 |
|
0.320 |
0.219 |
|
|
32.0% |
21.9% |
|
|
White |
9,542 |
32,975 |
|
0.381 |
0.545 |
|
|
38.1% |
54.5% |
|
|
All Other |
3,260 |
3,978 |
|
0.130 |
0.066 |
|
|
13.0% |
6.6% |
|
|
Total |
25,071 |
60,522 |
|
1.000 |
1.001 |
|
|
100% |
100% |
What general trends and conclusions do you make from these frequency distributions?
- In a study of unequal health service outcomes, researchers at a large hospital system gathered data on patients from the previous year that had reported trouble with alcoholism. The patients for the study were chosen based on the severity of their condition. That is, all the patients in the group should have received roughly the same diagnosis and treatment options from their physician. The cross classified frequency distribution in the table below classifies each of the patients by race and ethnicity and the type of treatment the attending physician suggested to the patient.
|
Race |
Treatment Option |
|||
|
Or Ethnicity |
None |
Outpatient |
Inpatient |
Total |
|
African American |
1001 |
631 |
315 |
1947 |
|
Asian |
53 |
122 |
285 |
460 |
|
Hispanic |
1507 |
732 |
744 |
2983 |
|
Native American |
112 |
58 |
52 |
222 |
|
White |
983 |
1203 |
2096 |
4237 |
|
Total |
3611 |
2746 |
3492 |
9849 |
Compute the relative frequencies and percentages for this frequency distribution. What conclusions can be drawn from this table?