Interval half width

LCL and UPL
➢
14.5 – TESTING THE SIGNIFICANCE OF THE SLOPE OF THE
REGRESSION EQUATION
o
Interpreting the Meaning of Slope = 0
▪
For any value of x, the predicted value of y will always equal the
yintercept
▪
No relationship b/w the dependent and independent variables
▪
Perform a hypothesis test to det. if the population regression slope
(M) is significantly diff from zero based on the sample regression
slope (m)
➢
The Hypothesis Test
o
Step 1: The hypotheses:

H0: M = 0

H1: M ≠ 0
▪
If reject null hypothesis

There is a significant relationship b/w the dependent and
independent variables
o
Step 2: T Test Statistic
▪
T = (m – M) / Sb

m = sample regression slope

M = population regression slope (from null hypothesis)

Sb = standard error of the slope
▪
In our example = 2.96
o
Step 3. Standard Error of the Slope
(Sb) – measures how consistent the
slope of the regression equation (b) would be if several sets of samples
from the population were selected and the regression equation were
derived for each of them
Sb = (Se) ÷ √ ∑x² –
n(sample mean of x)²
▪
Regression equation calculated for many diff samples
▪
Det. how close the slopes for these samples are to one another
▪
Small Sb – increases likelihood that can est. significant
relationship b/w the two variables
▪
Large Sb  low likelihood that can est. significant relationship b/w
the two variables
▪
In our example = 1.41
o
Step 4. Critical TScore
▪
DF = (n – 2) = 4
▪
∞ = 0.05
▪
Twotail
▪
→ 2.776
o
Step 5. Comparing T and the TScore
▪
2.96 (T) > 2.776 (TScore) → REJECT
o
Step 6: State your conslusions
▪
Reject null hypothesis
▪
The population regression slope (M) is NOT equal to zero
▪
A relationship exists b/w the hours studied and exam grades
CONFIDENCE Interval for M
o
Formula – Confidence Interval for the Regression Slope
▪
CI = m ± (tscore • Sb)
▪
Critical TScore

∞ = 0.05 of 95% condience interval at DF (n –
2) = 4

→ 2.776
▪
In our example, CI = 4.1786 ± 3.914

UCL = 8.093

LCL = 0.265
o
Results:
▪
95% confident that true population slope is b/w

0.265 and 8.093
▪
Every additional hour of study (one more x) will increase your
score by 0.265 – 8.093 points
▪
Confidence interval does NOT include zero

Evidence to conclude that there is a relationship b/w the
variables
o
Finding it in Excel
▪
Se = under Regression Statistics
▪
TScore in “t Stat” column, 2
nd
row
▪
LCL and UCL in “lower 95%” column, 2
nd
row
▪
Pvalue in Pvalue column, 2
nd
row
o
PValue in Excel
▪
Represents area to the right and left of the extreme ± values of the
TScore
▪
If P < ∞ → REJECT
▪
When pvalue and ttest are identical

b/c only have one independent variable
➢
14.6 – ASSUMPTIONS FOR REGRESSION ANALYSIS
o
The Basics
▪
Key assumptions need to hold true for regression analysis results to
be reliable
o
Assumption: the relationship b/w the independent and dependent variables
is linear
▪
Nonlinear pattern affects regression results

Some predicted Y values may end up in negative
o
Assumption 1: Residuals
▪
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 Fall '12
 Donnelly
 Normal Distribution, Null hypothesis, Hypothesis testing, Statistical hypothesis testing