Slides are available at: www.msu.edu/~whitery2/mkt317slides.html

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

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

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

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

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

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

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)

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

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

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

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)

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

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

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

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

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

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

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

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

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

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.

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