Example 1.
The fifth-grade students in Mr. Brown’s class were given a vocabulary test and a
reading comprehension test.
TASKS
1] Create a scatterplot.
Plot the
scores on two variables for each subject.
2]
Compute the correlation coefficient
I.
Scatterplot
SPSS for Windows
A. Define
the variables: voc and com. Enter data.
Note that
you may press the Tab key to move one cell to the right (the second
variable).
B. Create
a scatterplot with scores on the vocabulary test as the X variable.
a. From the menus choose: Graphs / Scatter
Select the Simple picture button.
Select `com` as the Y Axis variable.
b. Modify the chart to produce a linear regression line.
Instructions for
(a) Double-click on the scatterplot to bring up the Chart Editor window.
(b) From the Chart Editor Window menus choose: Chart / Options
In the Fit Line area, click
on Total. Click on the Fit Options button.
Fit Method. Select Linear Regression.
Regression Options. Click on Display R-squared in legend as shown above.
Click Continue and OK. Close the Chart Editor window and return to the viewer window.
(a) Double-click on the scatterplot to bring up the Chart Editor window.
(b) Click on any one of the data points to highlight all of them.
(c) From the Chart Editor Window menus choose: Chart / Add Chart Element/ Fit
Line at Total
SPSS
Output
A.
Inspect the scatterplot.
a. It is a positive linear relationship. There is a tendency for high scores on the vocabulary test to be associated with high scores on the reading comprehension test. There is a tendency for low scores on the vocabulary test to be associated with low scores on the reading comprehension test
B.
R-squared = .6378
About 64 percent of the variance in the reading comprehension scores is predictable from the vocabulary scores and vice versa.
II. Apply the Bivariate procedure
and obtain the Pearson r.
SPSS
for Windows
A. From
the menus choose: Analyze / Correlate / Bivariate
B. Select
the variables to be correlated (voc and com). Click OK to obtain the default
Pearson correlation.
SPSS Printout
The Pearson product-moment correlation coefficient is about .80.
Note that the degrees of freedom are 7 - 2 = 5.Test of Pearson Correlation
r = .80
Research Question: Is there a
significant relationship between the vocabulary test scores and the reading
comprehension test scores?
Hypotheses
The null
hypothesis states that the population correlation, rho, is
zero.
The alternative hypothesis states that the population correlation, rho, is
different from zero.
To test if the observed correlation is significantly different from zero, first compute a test statistic, a t ratio. How far does the sample correlation (r) from the assumed population correlation (0) in standard error units?
The t ratio can be computed as
The denominator is the standard
error of r.
Examine the formula
Note that as the sample size (N) increases, the standard error of r decreases. Even small correlations may be statistically significance.
Next, locate the obtained t value in the t distribution with 5 degrees of freedom and find the associated probability. Finally, compare the probability to the .05 significance level. Note that the two-tailed observed significance level was 031. Since the probability is less than .05, the researcher would reject the null hypothesis and declare the result to be statistically significant.
Report the results
The correlation between vocabulary and reading
comprehension was significant, r(5) = .80, p < .05.
Students
who earned high scores on the vocabulary test also tended to earn high scores on
the reading comprehension test. Students who earned low scores on the vocabulary
test also tended to earn low scores on the reading comprehension test.
Example 2.
Open an existing SPSS data file -- Cars. Create a scatterplot of miles per
gallon and vehicle weight. Compute the correlation between the two variables.
A. Open
the Cars data file.
B. Create
a scatterplot: vehicle weight ( X) and miles per gallon (Y)
a. From the menus choose: Graphs / Scatter
Select the Simple picture button. Click on Define.
Select ` Miles per Gallon ` as the Y Axis variable.
Select ` Vehicle Weight ` as the X Axis variable. Click on OK.
b. Modify the chart to produce a linear regression line.
(a) Double-click on the scatterplot to bring up the Chart Editor window.
(b) From the Chart Editor Window menus choose: Chart / Options
In the Fit Line area, click on Total. Click on the Fit Options button.
Fit Method. Select Linear Regression.
Regression Options. Click on Display R-squared in legend.
Click Continue. Click OK. Close the Chart Editor window.
(a) Double-click on the scatterplot to bring up the Chart Editor window.
(b) Click on any one of the data points to highlight all of them.
(c) From the Chart Editor Window menus choose: Chart / Add Chart Element/ Fit
Line at Total
What do you observe?
There is
a negative relationship between miles per gallon and vehicle weight.. About 65
percent of the variance in miles per gallon is predictable from vehicle weight.
C. From
the menus choose: Analyze / Correlate / Bivariate
Select
the variables to be correlated (Miles per Gallon and Vehicle Weight). Click OK
to obtain the default Pearson correlation. r = -.807
Example 3.
Add a categorical variable (Country of Origin) to the scatterplot. Create a
scatterplot of miles per gallon and vehicle weight by country of origin.
From the menus choose: Graphs / Scatter
Select the Simple picture button. Click on Define.
Select `Miles per Gallon` as the Y Axis variable.
Select `Vehicle Weight` as the X Axis variable.
Set Markers by: Select Country of Origin. Click on OK.
