14.10: Correlation Exercises
- Page ID
- 18130
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise \(\PageIndex{1}\)
What does a correlation assess?
- Answer
-
Correlations assess the strength and direction of a linear relationship between two quantitative variables.
Exercise \(\PageIndex{2}\)
What sort of relation is displayed in the scatterplot in Figure \(\PageIndex{1}\) below?
- Answer
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Strong, positive, linear relation
Exercise \(\PageIndex{3}\)
Create a scatterplot from the following data:
| Hours Studying | Overall Class Performance |
|---|---|
| 0.62 | 2.02 |
| 1.50 | 4.62 |
| 0.34 | 2.60 |
| 0.97 | 1.59 |
| 3.54 | 4.67 |
| 0.69 | 2.52 |
| 1.53 | 2.28 |
| 0.32 | 1.68 |
| 1.94 | 2.50 |
| 1.25 | 4.04 |
| 1.42 | 2.63 |
| 3.07 | 3.53 |
| 3.99 | 3.90 |
| 1.73 | 2.75 |
| 1.9 | 2.95 |
- Answer
-
Figure \(\PageIndex{2}\): Scatterplot of data from Table \(\PageIndex{1}\). (CC-BY-NC-SA Foster et al. from An Introduction to Psychological Statistics)
Exercise \(\PageIndex{4}\)
Using the data from Table \(\PageIndex{1}\), test for a statistically significant relation between the variables.
- Answer
-
Step 1:
Research Hypothesis: “There is a positive linear relation between time spent studying and overall performance in class.”
Null Hypothesis “There is no liner relation between time spent studying and overall performance in class.”
Step 2: \(df = 15 – 2 = 13, \alpha = 0.05, r_{Crit} = 0.514\).
Step 3: Using the Sum of Products table, you should find: \(\overline{X_{HS}} = 1.65, \overline{X_{OCP}}= 2.95, r = 0.65\). [Your means might differ due to rounding differences.
Step 4: Obtained statistic is greater than critical value, reject \(H_0\). There is a statistically significant, strong, positive relation between time spent studying (\(\overline{X_{HS}} = 1.65) and performance in class (\overline{X_{OCP}}= 2.95), \(r(13) = 0.65, p < .05\). This supports the research hypothesis.
Contributors and Attributions
Foster et al. (University of Missouri-St. Louis, Rice University, & University of Houston, Downtown Campus)