1. Slides are available at:

2. TOPICS COVERED Type I Error / Type II Error Multiple Regression
Levels of Measurement (NOIR)
Test of Two Means (t or z)
Paired Samples (t test)
Two Proportions (z test)
F–test
One Way ANOVA
Tukey test
Two Way Anova w/ Interactions
Simple Regression
Correlation
Multiple Regression
Adjusted R2
Dummy Variables
Time Series
Autoregressive Model
Moving (Centered) Average
Seasonal Index
Chi-Square Goodness of Fit
Chi-Square Test of Independence
*p value

3. Error Types
Type 1:
Rejecting a True H0
Type 2:
Accepting a False H0

4. Levels of Measurement (NOIR)
Nominal: IDENTIFY
Ordinal: ORDER
Interval: COMPARE INTERVALS
Ratio: COMPARE ABSOLUTE MAGNITUDES

5. Test of Two Means (t or z)
Criterion: n > 30 for both: z-Test
n < 30 for either: t-Test
H0: μ1 – μ2 = ≤ ≥ 0 H1: μ1 – μ2 ≠ < > 0
df = n1 + n2 - 2
T FORMULA
Z FORMULA

6. Paired Samples (t test)
How to know a paired sample test:
Average difference or D-BAR given
H0: μD = ≤ ≥ 0 H1: μD ≠ < > 0
df = n-1

7. Two Proportions (z test)
D0 = ?????
D0 = 0
POOL
D0 ≠ 0
NOT POOL
H0: p1 – p2 = ≤ ≥ 0
H1: p1 – p2 ≠ < > 0

8. F–test Test of Equality of Variance (σ2)
H0: σ12 = ≤ ≥ σ22 H1: σ12 ≠ < > σ22
Dfnum = (n of sample in numerator – 1)
Dfdenom = (n of sample in denominator – 1)

9. One Way ANOVA n = total sample size r = number of groups
SOURCE OF VARIATION DF
Sum of Squares
Mean Sum of Squares
F CALC
BETWEEN
(TREATMENT)
r-1
SSTR
SSTR / r-1
MSTR / MSE
WITHIN
(ERROR)
n-r
SSE
SSE / n-r
TOTAL
n-1
SST
SST / n-1
n = total sample size
r = number of groups
H0: μ1 = μ2 = … μr
H1: not all means equal
df = r-1, n-r

10. Tukey test H0: μi = μj for each pair of means
Which means are different?
H0: μi = μj for each pair of means
H1: μi ≠ μj for at least one pair of means
Reject H0 if
r = number of groups
n = total sample size

11. Two Way ANOVA w/Interactions
Effects on means due to:
factor a (αi )
factor b (βj )
and interaction between factor a and factor b
(αβij)
H0: αi = 0, for all i=1 to α
H1: not all αi = 0
H0: βj = 0, for all j=1 to β
H1: not all βj = 0
H0: α βij = 0, for all i and j
H1: not all αβij = 0

12. Two Way ANOVA w/Interactions (cont.)
SOURCE OF VARIATION DF
Sum of Squares
Mean Sum of Squares
F CALC
Factor a
a-1
SSA
SSA/a-1
MSA /
MSE
Factor b
b-1
SSB
SSB / b-1
MSB /
Interaction
(a-1)(b-1)
SSAB
SSAB/
MSAB /
Error
ab(n-1)
SSE
SSE/ab(n-1)
TOTAL
abn-1
SST
SST / abn-1
a= number of levels of a
b= number of levels of b
n= sample size per cell (combination of a and b)

13. Simple Regression Parameter estimates Coefficient of determination
Overall model test
H0: β1=0 H1: β1≠0
Individual parameters test
Confidence intervals

14. Correlation
H0: ρ = 0
H1: ρ ≠ 0
-1 = Strong negative (inverse) relationship
0 = no linear relationship
+1 = strong positive (direct) relationship

15. Multiple Regression Overall model test H0: β1=… βi=0
H1: at least one βi≠0
Individual parameters test
H0: β1=0 H1: β1≠0
H0: β2=0 H1: β2≠0
Coefficient of Multiple Determination
Adjusted R2
Confidence intervals

16. Dummy Variables Example
Four Seasons:
Winter, Spring, Summer, Fall
SEASON X1 X2 X3
WINTER
SPRING 1
SUMMER
FALL

17. TIME SERIES YEAR SALES (in thousands) t 1983 100.6 1984 102.9 1 1985
1984
102.9 1
1985
108.7 2
1986
128.4 3
1987
150.7 4
1988
149.6 5
1989
166.0 6
1990
161.6 7
1991
150.6 8
1992
174.0 9

18. YT SALES (in thousands)
Autoregressive Model
YEAR
YT SALES (in thousands)
YT-4
1983
100.6
1984
102.9
1985
108.7
1986
128.4
1987
150.7
1988
149.6
1989
166.0
1990
161.6
1991
150.6
1992
174.0

19. Moving Average and Centered Moving Average
TIME
SALES (1,000s)
MOVING AVERAGE
CENTERED
RATIO TO CENTERED MOVING AVERAGE
2001-1
4.8
2001-2
4.1
5.35
2001-3
6.0
5.475
1.10
5.6
2001-4
6.5
5.7375
1.13
5.875
2002-1
5.8
5.975
0.97
6.075
2002-2
5.2
6.1875
0.84
6.3
2002-3
6.8
2002-4
7.4

20. Seasonal Index
Average of Ratio to Centered Moving Average for same time periods
Given a seasonal index of .932 adjusted the predicted sales value of 5.03:
5.03 * .932 = 4.69

21. Chi-Square Goodness of Fit
K = number of categories
Oi = the observed frequency
Ei = the expected frequency
df = K-1
Ei = n*pi
n = sample size
pi = appropriate probability from the null hypothesis
Proportions are the same:
H0: p1 = p2 = p3 = …pk
H1: at least one proportion is not equal
Proportions are not all the same:
H0: p1 = p1H0 ... pk = pkH0

22. Chi-Square Test of Independence
Oij = observed value for cell ij
Ei = expected value for cell ij
df = (r-1)(c-1)
Eij = (ri*cj)/n
n = sample size
H0: The two classification variables are independent.
H1: The two classification variables are not independent.

23. P-Value p value < alpha: REJECT H0
p value > alpha: DO NOT REJECT H0