10.3.1: Exercises
- Page ID
- 49077
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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 Pew Research Center studies many different groups in the United States. One of the center’s projects is the Pew Internet and American Life Project. In this project, the research center learns how people in the United States use computers and technology.
In one study, researchers asked people, “Do you use a computer at your workplace, at school, at home, or anywhere else on at least an occasional basis?” The possible responses to this question were “Yes” and “No”. Researchers also recorded information about each respondent’s urbanity, that is whether the respondent lived in an “Urban” area (a city), a “Suburban” area (a neighborhood outside city limits), or a “Rural” area (not in a neighborhood).
Researchers obtained the following results, based on a sample of 8,296 individuals:
|
Urbanity |
||||
|
Urban |
Suburban |
Rural |
||
|
Response |
Yes |
1946 |
3533 |
943 |
|
(“Do you use a computer?”) |
No |
537 |
835 |
502 |
Do these data support the claim that there is a relationship between a person’s response to the question about computer use and the person’s urbanity? Execute a complete chi-square test for independence for this case. Use a significance level of \(\alpha=0.01\).
- Step 1: What are the appropriate hypotheses for this test?
- Step 2: Collect the Data
The table below displays the row, column and grand totals, and the expected frequencies for all but one cell. Compute and enter in the missing expected frequency. Round the value to two decimal places.
Computer Usage
Urban
Suburban
Rural
Totals
Yes
3381.30
1118.59
6422
No
560.89
986.70
326.41
1874
Totals 2483 4368 1445 8296
- Step 3: Assess the Evidence
- Each pair of observed and expected frequencies are provided in the table below. Compute the missing contribution to the \(\chi^2\) test statistic. Round the values to two decimal places.
Pairings of Values
O = Observed Frequency
E = Expected Frequency
\(\frac{(O-E)^2}{E}\)
Urban / Yes
1946
Urban / No
537
560.89
1.02
Suburban / Yes
3533
3381.30
6.81
Suburban / No
835
986.70
23.32
Rural / Yes
943
1118.59
27.56
Rural / No
502
326.41
94.46
- All expected frequencies are greater than 5, so we can proceed with the hypothesis test. What is the value of the \(\chi^2\) test statistic? Write the value to two decimal places.
- Use the desmos graph https://www.desmos.com/calculator/bjohldwaym to determine the P-value.
- Each pair of observed and expected frequencies are provided in the table below. Compute the missing contribution to the \(\chi^2\) test statistic. Round the values to two decimal places.
- Step 4: Make a Decision
At the 1% significance level, write an appropriate conclusion.