11.3 Test of Independence
- Page ID
- 36533
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Section 11.3 Test of Independence
Learning Objective:
In this section, you will:
• Conduct and interpret chi-square test of independence hypothesis tests
A test of independence is used to determine whether two factors are independent or not. We use the graphing calculator, 𝑪:𝝌𝟐 − 𝑻𝒆𝒔𝒕, with test statistic 𝜒, with
𝐸 df = (# of rows – 1)(# of columns – 1), O = Observed values, E = expected values
Enter data in a matrix [A] using a graphing calculator, Matrix (2nd x-1), Edit, input number of rows and columns. Note that the 𝝌𝟐 − 𝑻𝒆𝒔𝒕 will create a matrix of the expected values and place it in matrix [B].
Example 1: In a volunteer group, adults 21 and older volunteer from one to nine hours each week to spend time with a disabled senior citizen. The program recruits among community college students, four-year college students, and nonstudents. In the table below is a sample of the adult volunteers and the number of hours they volunteer per week.
|
Type of Volunteer |
1-3 Hours |
4-6 Hours |
7-9 Hours |
|
Community College Student |
111 |
96 |
48 |
|
Four-Year College Student |
96 |
133 |
61 |
|
Nonstudents |
91 |
150 |
53 |
Is the number of hours volunteered independent of the type of volunteer? Test at a 5% significance level.
- Null and Alternative Hypothesis
- Calculator Work
- Test Statistic and P-Value
- Conclusion about the null hypothesis
- Final conclusion that addresses the original claim
Notes 11.3
Example 2: De Anza College is interested in the relationship between anxiety level and the need to succeed in school. A random sample of 400 students took a test that measured anxiety level and need to succeed in school. The table below shows the results. De Anza College wants to know if anxiety level and need to succeed in school are independent events. Test at a 5% significance level.
|
Need to Succeed in School |
High Anxiety |
Med-high Anxiety |
Medium Anxiety |
Med-low Anxiety |
Low Anxiety |
|
High Need |
35 |
42 |
53 |
15 |
10 |
|
Medium Need |
18 |
48 |
63 |
33 |
31 |
|
Low Need |
4 |
5 |
11 |
15 |
17 |
- Null and Alternative Hypothesis
- Calculator Work
- Test Statistic and P-Value
- Conclusion about the null hypothesis
- Final conclusion that addresses the original claim
For more information and examples see online textbook OpenStax Introductory Statistics pages 633-638.
“Introduction to Statistics” by OpenStax, used is licensed under a Creative Commons Attribution