Correlation and Simple Linear Regression
Using the Excel SUMMARY OUTPUT below, complete the following:
All calculations accurate to four decimal places
1.) What is the value of the correlation coefficient?
2.) What is the value of the coefficient of determination?
3.) What is the value of the y-intercept of the least squares line?
4.) What is the value of the slope of the least squares line?
5.) Write the equation of the least squares line for this set of data.
6.) Is the slope of the line significant (yes or no):
a.) at p < .05?
b.) at p < .01?
c.) at p < .001?
7.) Based on these results, write a sentence explaining the relationship between the dependent and independent variable.
8.) Compute predicted y for x = 10 using the equation of the least squares line for this data set.
9.) What percent of the variation in y (dependent variable) is explained by x (independent) variable?
10.) What value represents the strength of the relationship between the independent and dependent variable?
In simple linear regression, how many independent variables may be used?
How many dependent variables may be used?
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.95624857
R Square 0.914411327
Adjusted R Square 0.897293593
Standard Error 5.089839735
Observations 7
ANOVA
df SS MS F Significance F
Regression 1 1383.8962 1383.8962 53.4189 0.00075
Residual 5 129.53234 25.906469
Total 6
Coefficients Standard Error t Stat P-value
Intercept -3.88811 4.23842 -0.91751 0.40096
X Variable 1 5.81993 0.79659 7.30883 0.00075
please see the attached file
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SOLUTION This solution is FREE courtesy of BrainMass!
See attached file
Using the Excel SUMMARY OUTPUT below, complete the following:
All calculations accurate to four decimal places
1.) What is the value of the correlation coefficient?
Regression Statistics
Multiple R 0.95624857
R Square 0.914411327
Adjusted R Square 0.897293593
Standard Error 5.089839735
Observations 7
R Square = 0.914411327
Therefore r= correlation coefficient=√ R Square= √0.914411327 =0.956248
Answer: r= correlation coefficient=0.956248
2.) What is the value of the coefficient of determination?
Regression Statistics
Multiple R 0.95624857
R Square 0.914411327
Adjusted R Square 0.897293593
Standard Error 5.089839735
Observations 7
coefficient of determination= R Square = 0.914411327
Answer: 0.914411327
3.) What is the value of the y-intercept of the least squares line?
Coefficients Standard Error t Stat P-value
Intercept -3.88811 4.23842 -0.91751 0.40096
X Variable 1 5.81993 0.79659 7.30883 0.00075
Answer: -3.88811
4.) What is the value of the slope of the least squares line?
Coefficients Standard Error t Stat P-value
Intercept -3.88811 4.23842 -0.91751 0.40096
X Variable 1 5.81993 0.79659 7.30883 0.00075
Answer: 5.81993
5.) Write the equation of the least squares line for this set of data.
Coefficients Standard Error t Stat P-value
Intercept -3.88811 4.23842 -0.91751 0.40096
X Variable 1 5.81993 0.79659 7.30883 0.00075
Y=-3.88811 + 5.81993 X
6.) Is the slope of the line significant (yes or no):
a.) at p < .05?
b.) at p < .01?
c.) at p < .001?
Coefficients Standard Error t Stat P-value
Intercept -3.88811 4.23842 -0.91751 0.40096
X Variable 1 5.81993 0.79659 7.30883 0.00075
p value=0.00075
a.) at p < .05? p = .05 >0.00075 hence significant
b.) at p < .01? p = .01 >0.00075 hence significant
c.) at p < .001? p = .001 >0.00075 hence significant
7.) Based on these results, write a sentence explaining the relationship between the dependent and independent variable.
For a unit change in the dependent variable X the dependent variable Y changes by 5.81993 (slope of regression line)
8.) Compute predicted y for x = 10 using the equation of the least squares line for this data set.
Y=-3.88811 + 5.81993 X
X= 10
Therefore Y= -3.88811 + 5.81993 X = -3.88811 + 5.81993 * 10= -3.88811 + 58.1993 = 54.31119
Answer : 54.31119
9.) What percent of the variation in y (dependent variable) is explain by x (independent) variable?
Regression Statistics
Multiple R 0.95624857
R Square 0.914411327
Adjusted R Square 0.897293593
Standard Error 5.089839735
Observations 7
R Square 0.914411327
percent of the variation in y (dependent variable) is explain by x (independent) variable= 91.44% ( R square value expressed as a percentage)
10.) What value represents the strength of the relationship between the independent and dependent variable?
ANOVA
df SS MS F Significance F
Regression 1 1383.8962 1383.8962 53.4189 0.00075
Residual 5 129.53234 25.906469
Total 6
F value and Significance F
F value is very high=53.4189
Corresponding to this value of F, Significance F =0.00075 Which means that the relationship is significant at level=0.00075 or 0.075%
This means that the strength of the relationship between the independent and dependent variable is very high ( the lower the level of significance the higher the strength of relationship)
In simple linear regression, how many independent variables may be used?
Any number
How many dependent variables may be used?
One
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.95624857
R Square 0.914411327
Adjusted R Square 0.897293593
Standard Error 5.089839735
Observations 7
ANOVA
df SS MS F Significance F
Regression 1 1383.8962 1383.8962 53.4189 0.00075
Residual 5 129.53234 25.906469
Total 6
Coefficients Standard Error t Stat P-value
Intercept -3.88811 4.23842 -0.91751 0.40096
X Variable 1 5.81993 0.79659 7.30883 0.00075
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