SST in Excel
▪
Column 1 = xij

Shows each data value in a vertical column
▪
Column 2 =
x
x

Shows the same grand mean in every column
▪
Column 3 = (xij –
)
x
x

(Column 1 – Column 2)
▪
Column 4 = (xij –
)²
x
x

Square the data points in column 3
▪
ANSWER = sum of data points in column 4
o
Mean Square Total
(MST) – another term for the variance of the sample
data
▪
FORMULA
▪
MST = (SST) ÷ (n – 1)
▪
In our example:

15 data points

SST = 833.73

MST = (833.73) ÷ (15 – 1) = 59.55
o
Notes:
▪
If null hypothesis is true (assumed unless proven otherwise)
▪
→ levels being compared are from same population
with:

Mean = µ

Variance = σ²
▪
THEN MST is a good estimate for the population variance σ²
●
Step 4: Partition SST the total sum of squares (SST) into the sum of squares b/w
(SSB) and the sum of squares w/in (SSW)
o
SST = divided into 2 components
▪
The sum of squares b/w SSB
▪
The sum of squares w/in SSW
▪
SST = SSB + SSW
o
Sum of squares between SSB
(SSB) – measures the variation between
each sample mean and the grand mean of the data
▪
(sample mean – grand mean)² • (number of data points in sample)
▪
Find this for each sample population
▪
Add up the results for → SSB
o
Mean Square Between MSB
(MSB) – provides an estimate for the
population variance (σ²) if the null hypothesis is true
▪
(SSB) ÷ (k – 1)
▪
Where k = number of populations being compared

Ex: if 4 types of smartphones being compared, k = 4
o
Sum of Squares within SSW
(SSW) – measures the variation between
each data value and the corresponding sample mean
▪
SSW = (SST – SSB)
o
Mean Square within MSW
(MSW) – provides another estimate for the
population variance (σ²)
▪
(SSW) ÷ (n – k)

n = number of observations (total data points)

k = number of populations being observed (number of
sample populations)
●
Step 5: Calculate the appropriate test statistic
▪
Use the Ftest statistic for oneway ANOVA

Ronald Fisher = statistician
o
Formula – Ftest Statistic for OneWay ANOVA
▪
= (MSB) ÷ (MSW
▪
One way ANOVA equations = summarized on page 519
o
IF null hypothesis was true
▪
MSB and MSW would be close to one another
▪
Critical Fscore would be close to 1.0
o
IF null hypothesis is NOT true
▪
Expect MSB to be larger than MSW
▪
Critical Fscore exceeds 1.0
●
Step 6: Determine the appropriate critical value
o
Fdistribution
▪
Righskewed
▪
Rejection region in right tail
▪
ALWAYS onetail (upper) hypothesis test
▪
Critical value (F∞) = found in Appendix A
o
2 types of Degrees of Freedom
▪
D1 & D2
▪
Correspond to degrees of freedom for:

Sum of squares b/w SSB

Sum of squares w/in SSW
▪
D1 (COLUMN) = k – 1

Ex: (4 – 1) = 3
▪
D2 (ROW) = n – k

Ex: (15 – 4) = 11
▪
Column 3 / Row 11, where ∞ = 0.05

Critical value of 3.587
●
Step 7: compare the Ftest statistic (F) with the critical (Fscore)
o
Decision Rules – ANOVA
▪
F ≤ Fscore → DO NOT REJECT H0
▪
F > Fscore → REJECT H0
o
In our example:
▪
Fscore = 3.587 and F = 9.86
▪
F > Fscore → REJECT
●
Step 8: State your conclusions
o
Reject the null hypothesis
▪
Conclude that not all 4 population means are equal (some
difference)
▪
NOT enough info to decide which populations have significantly
different averages (find this out later)
o
ANOVA = compares 2 types of variances
▪
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 Fall '12
 Donnelly
 Normal Distribution, Null hypothesis, Hypothesis testing, Statistical hypothesis testing