With a small sample, a single point can have a large effect on the magnitude of the correlation. To create the following data, we started with the scores from problem 8 and changed the first X value from X = 1 to X = 6.
X | Y |
6 | 6 |
4 | 1 |
1 | 4 |
1 | 3 |
3 | 1 |
a. Sketch a scatter plot and estimate the value of the Pearson correlation.
b. Compute the Pearson correlation.
a.
To Construct: A scatter plot.
To determine: The estimated value of the Pearson correlation.
Output using the SPSS software is given below:
The estimated value of the Pearson correlation is around 0.25 to 0.30.
Given info:
The scores of X and Y.
Software procedure:
Step-by-step procedure to obtain the scatter plot using the SPSS software:
Justification:
The scatter plot shows points clustered around a line sloping up to the right.
Thus, the estimated value of the Pearson correlation, just by looking at the scatter plot, is around 0.25 to 0.30.
b.
To Calculate: The value of Pearson correlation.
The Pearson correlation is 0.277.
Calculation:
The Pearson correlation (r) is calculated as:
Here SP is calculated as:
Squared deviations SS_{x} and SS_{y} are calculated as:
The below table showing the calculations required for calculating correlation:
S.No. | Scores | Deviations | squared deviation | Products | |||
X | Y | (X-M_{x}) | (Y-M_{y}) | (X-M_{x})^{2} | (Y-M_{y})^{2} | (X-M_{x})(Y-M_{y}) | |
1 | 6 | 6 | 3 | 3 | 9 | 9 | 9 |
2 | 4 | 1 | 1 | -2 | 1 | 4 | -2 |
3 | 1 | 4 | -2 | 1 | 4 | 1 | -2 |
4 | 1 | 3 | -2 | 0 | 4 | 0 | 0 |
5 | 3 | 1 | 0 | -2 | 0 | 4 | 0 |
Sum | 3 | 3 | 18 | 18 | 5 |
So the sum of product of deviation (SP) is 5, SS_{x} is 18 and SS_{y} is 18, substitute these values in Pearson correlation formula then:
Thus, the Pearson correlation is 0.277.
Essentials of Statistics for The Behavioral Sciences (MindTap Course List)
Probability and Statistics for Engineering and the Sciences
Understanding Basic Statistics
Essentials Of Statistics
Statistics for The Behavioral Sciences (MindTap Course List)
Mathematical Excursions (MindTap Course List)
Single Variable Calculus
Mathematical Applications for the Management, Life, and Social Sciences
Calculus of a Single Variable
Elements Of Modern Algebra
Finite Mathematics
Multivariable Calculus
Calculus: Early Transcendental Functions (MindTap Course List)
Calculus: Early Transcendental Functions
Trigonometry (MindTap Course List)
Contemporary Mathematics for Business & Consumers
Single Variable Calculus: Early Transcendentals, Volume I
Calculus: Early Transcendentals
Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach
Applied Calculus
Finite Mathematics and Applied Calculus (MindTap Course List)
Elementary Technical Mathematics
Calculus: An Applied Approach (MindTap Course List)
Single Variable Calculus: Early Transcendentals
Calculus (MindTap Course List)
Intermediate Algebra
Finite Mathematics for the Managerial, Life, and Social Sciences
Precalculus: Mathematics for Calculus (Standalone Book)
Elementary Geometry For College Students, 7e
Introduction To Statistics And Data Analysis
Elementary Geometry for College Students
Study Guide for Stewart's Multivariable Calculus, 8th
Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th
Find all the answers to your study problems with bartleby.
Textbook solutions plus Q&A. Get As ASAP arrow_forward