Measures the squared diff b/w each data point and the average of
the data points

Larger difference b/w data points = larger SST
o
Divide SST → SSR and SSE
▪
SST = SSR + SSE
o
Sum of Squares Error
(SSE) – measures the variation in the dependent
variable that is explained by variables other than the independent variable
▪
SSE = ∑y²  b∑y  m∑xy

Where b = yintercept

M = slope
▪
In our example → 74.62
o
Sum of Squares Regression
(SSR) – measures the amount of variation in
the dependent variable that is explained by the independent variable
▪
SSR = ∑(predicted value of y at given value of x –
average value of the dependent variable from the
sample)²
▪
SSR = SST – SSE
▪
In our example → 237.5
●
Calculating the Coefficient of Determination
o
Coefficient of Determination
(R²) – measures the percentage of the total
variation of our dependent variable that is explained by our independent
variable from a sample
▪
R² = SSR / SST
▪
R² = (r)²
▪
In our example → 0.686

Conclude that 68.6% of total variation can be explained by
the independent variable (hours of study)
o
The Basics
▪
Value ranges from 0 – 100%
▪
Higher values = more desirable

b/c want to explain most of variation by independent
variable

indicate stronger relationship b/w dependent and
independent variables
▪
low values may indicate:

using wrong independent variable

need additional independent variables to explain variation
in dependent variable
●
Conducting a Hypothesis Test to Determine the Significance of the
Coefficient of Determination
o
Population Coefficient of Determination
(p²) – measures for an entire
population the percentage of the total variation of a dependent variable
that is explained by an independent variable
▪
Unknown in our example
▪
Perform hypothesis test to det. if p² is significantly diff from zero
based on R²
▪
If reject null hypothesis

Have enough evidence from our sample to conclude that a
relationship does exist b/w the 2 variables
o
Step 1: Hypothesis Statements
▪
H0: p² ≤ 0
▪
H1: p² > 0
o
Step 2: Set level of significance
▪
Set ∞ = 0.05
o
Step 3: FTest Statistic
▪
F = SSR ÷ (SSE / n – 2)
▪
In our example → 8.73
o
Step 4: Find the critical FScore
▪
Identifies the rejection region for the hypothesis test
▪
Follows Fdistribution (Table 6, Appendix A)
▪
Degrees of Freedom

D1 = 1 (always one b/c only one independent variable)

D2 = (n – 2)
▪
In our example, ∞ = 0.05, D1 = 1, D2 = 4

→ 7.709
o
Step 5: Comparing the FTest Statistic (F) with the Critical (FScore)
▪
8.73 (F) > 7.709 (Fscore) → REJECT
o
Step 6: State your conclusions
▪
Conclude that the coefficient of determination (R²) is greater than
zero – appears to be a relationship b/w the two variables
o
In the REGRESSION OUTPUT – EXCEL
▪
Regression Statistics > R Square

= coefficient of determination, R²
▪
Sum of squares (ANOVA – SS column)

Residual = error sum of squares

Regression
▪
ANOVA F column = Fscore
▪
ANOVA DF shows degrees of freedom for

D1 (regression)

D2 (residual)
▪
ANOVA Significance F = pvalue

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 Fall '12
 Donnelly
 Normal Distribution, Null hypothesis, Hypothesis testing, Statistical hypothesis testing