12.E: Correlations (Exercises)
- Page ID
- 7166
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- What does a correlation assess?
- Answer:
-
Correlations assess the linear relation between two continuous variables
- What are the three characteristics of a correlation coefficient?
- What is the difference between covariance and correlation?
- Answer:
-
Covariance is an unstandardized measure of how related two continuous variables are. Correlations are standardized versions of covariance that fall between negative 1 and positive 1.
- Why is it important to visualize correlational data in a scatterplot before performing analyses?
- What sort of relation is displayed in the scatterplot below?
- Answer:
-
Strong, positive, linear relation
- What is the direction and magnitude of the following correlation coefficients
- -0.81
- 0.40
- 0.15
- -0.08
- 0.29
- 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:
-
Your scatterplot should look similar to this:
- In the following correlation matrix, what is the relation (number, direction, and magnitude) between…
- Pay and Satisfaction
- Stress and Health
| Workplace | Pay | Satisfaction | Stress | Health |
|---|---|---|---|---|
| Pay | 1.00 | |||
| Satisfaction | 0.68 | 1.00 | ||
| Stress | 0.02 | -0.23 | 1.00 | |
| Health | 0.05 | 0.15 | -0.48 | 1.00 |
- Using the data from problem 7, test for a statistically significant relation between the variables.
- Answer:
-
Step 1: \(H_0: ρ = 0\), “There is no relation between time spent studying and overall performance in class”, \(H_A: ρ > 0\), “There is a positive relation between time spent studying and overall performance in class.”
Step 2: \(df = 15 – 2 = 13, α = 0.05\), 1-tailed test, \(r^* = 0.441\).
Step 3: Using the Sum of Products table, you should find: \(\overline{X} = 1.61, SS_X = 17.44, \overline{Y}= 2.95, SS_Y = 13.60, SP = 10.06, r = 0.65\).
Step 4: Obtained statistic is greater than critical value, reject \(H_0\). There is a statistically significant, strong, positive relation between time spent studying and performance in class, \(r(13) = 0.65, p < .05\).
- A researcher collects data from 100 people to assess whether there is any relation between level of education and levels of civic engagement. The researcher finds the following descriptive values: \(\overline{X}= 4.02, s_x = 1.15, \overline{Y}= 15.92, s_y = 5.01, SS_X = 130.93, SS_Y = 2484.91, SP = 159.39\). Test for a significant relation using the four step hypothesis testing procedure.