Hypothesis Testing: Means and Proportions

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1 MBACATÓLICA JAN/APRIL 006 Marketing Research Fernando S. Machado Week 8 Hypothesis Testing: Means and Proportions Analysis of Variance: One way ANOVA Analysis of Variance: N-way ANOVA Hypothesis Testing: Means and Proportions One Sample T test T test for Independent Samples T test for Paired Samples

2 A Classification of Hypothesis Testing Procedures for Examining Differences Hypothesis Tests Parametric Tests (Metric Data) Non-parametric Tests (Nonmetric Data) One Sample * t test * Z test Independent Samples * Two-Group t test * Z test Two or More Samples Paired Samples * Paired t test One Sample * Chi-Square * K-S * Runs * Binomial Independent Samples * Chi-Square * Mann-Whitney * K-S * Median Two or More Samples Paired Samples * Sign * Wilcoxon * McNemar 3 Hypothesis Testing For Differences Between Means (Cont.) Hypothesis Testing Criteria Depends on Whether the samples are obtained from different or related populations Whether the population standard deviation is known or not known If the population standard deviation is not known, whether they can be assumed to be equal or not 4

3 Two Independent-Samples t Tests (internet usage by sex) S ummary S tatistics N umber S tandard of Cases Me an De viation Male Female F Test for Equa lity of V ariances F -tail value probability t Test Equal Varianc es Assumed Equ al Variances N ot Assu med t Degrees of -tail t Degrees of -tail value freedom probability value freedom probability Paired-Samples t Test Number Standard Standard Variable of Cases Mean Deviation Error Internet Attitude Technology Attitude Difference Internet - Technology Difference Standard Standard -tail t Degrees of -tail Mean deviation error Correlation prob. value freedom probability

4 Analysis of Variance: One Way Anova When to Use ANOVA ANOVA Basic Concepts Between and Within Group Variation 7 Similarities and Differences between ANOVA, Regression, and Log. Regression /Discr. Analysis ANOVA REGREION LOG. REG./DISC. ANALSIS Similarities Number of One One One dependent variables Number of independent Multiple Multiple Multiple variables Differences Nature of the dependent Metric Metric Categorical variable Nature of the independent Categorical Metric Metric variables 8 4

5 Analysis of Variance (ANOVA) Statistical technique for examining the diferences between two or more populations. Example: RCA Records has a million copies of Metallica s latest CD to allocate among 5 regional distribution centres. Data was collected on the past levels of sales from similar heavy-metal bands in each of the 5 regions. Are there any significant differences between those regions in terms of average sales of heavy metal CD s? Basic Concepts Factor: Categorical (non-metric) independent variable. Treatment: A particular combination of factor levels or categories One way (one-factor) Analysis of Variance: An ANOVA technique in which there is only one factor. 9 ANOVA and Causal Statistical Designs Completely randomized design: Three prices are under consideration for a new product: 39, 44 e 49 cents of euro. To determine the influence the various price levels will have on sales, three samples of supermarkets were randomly selected from the geographic area of interest. Are there any significant differences between the average levels of sales for the various price levels? EG: R X O EG: R X O EG3: R X 3 O 3 0 5

6 One Way (One-Factor) ANOVA Studies the effect of c treatments on one response variable Determine whether or not there are any statistically significant differences between the treatment means µ, µ,... µ c H o : all treatments have same effect on mean responses H : At least of µ, µ... µ c are different To Test Hypothesis, Compute the Ratio Between the "Between Treatment" Variance and "Within Treatment" Variance An Example Store Number Coupon Level In-Store Promotion Sales Clientel Rating

7 Testing for Differences between Means: Between-Group and Within-Group Variation High In-Store Medium In-Store Promotion (ISP) Promotion (ISP) Store Store Sales Number Number Sales 0,00 6 8,00 9,00 7 8,00 3 0,00 8 7,00 4 8,00 9 9,00 5 9,00 0 6,00 6 8,00 4,00 7 9,00 5,00 8 7,00 3 5,00 9 7,00 4 6,00 0 6,00 5 4,00 Average 8,30 Average 6,0 High In-Store Promotion (ISP) Store Number Sales Medium In-Store Promotion (ISP) Store Sales Number 8,30 6 6,0 8,30 7 6,0 3 8,30 8 6,0 4 8,30 9 6,0 5 8,30 0 6,0 6 8,30 6,0 7 8,30 6,0 8 8,30 3 6,0 9 8,30 4 6,0 0 8,30 5 6,0 Average 8,30 Average 6,0 3 Decomposition of the Total Variation: One-Way ANOVA Within Category Variation within Category Mean j n i ij n Independent Variable Categories X X X X3 Xc 3 c 3 c : : : : n n n3 nc N 3 c Between Category Variation between Total Sample c Total Variation y Grand Mean n j i ij N 4 7

8 Decomposition of the total variation of within y between + within error y N ( i ) i c n ( ij j ) j i η y X error y between Measure of strength of effect of X on : x c n( j ) j 5 Test of equality of means H 0 : µ µ µ 3...µ c MS x MS between x ( c ) MS error error MSwithin ( N c) F Statistic: X ( c ) F error ( N c ) MS MS X error df(c-), (N-c) Pvalue: Probability that the F-ratio would be larger than the calculated F-ratio, given the null hypothesis 6 8

9 Effect of In-Store Promotion on Sales Store Level of In-Store Promotion Number High Medium Low Column totals Category means: 83/0 6/0 37/0 j Grand mean: ( )/ One-Way ANOVA: Effect of In-store Promotion on Store Sales Source of Sum of df Mean F ratio F prob. Variation squares square Between groups (Promotion) Within groups (Error) TOTAL Cell means Level of Count Mean Promotion High () Medium () Low (3) TOTAL

10 Analysis of Variance: N- Way Anova When to use N-Way ANOVA Interaction Effects 9 N-Way ANOVA An ANOVA model in which two or more factors are involved. Sometimes we are interested in assessing the effect of more than one factor on a single dependent variable. Ex: What is the effect of sex and age on the attitudes towards our brand? Ex: How do consumers intentions to buy a brand vary with different levels of price and different levels of advertising? In particular, how do the effects of price and advertising interact with each other? Assume that we have 3 levels of advertising and levels of price. Then a (full) factorial design has 6 (3*) treatments. 0 0

11 Interaction Effect Impact of one treatment will not be the same for each condition of the other treatment Hypothesis of no interaction can be tested using F-ratio for interaction Decomposition of total variation of y X + X + XX + error multiple η X + X + XX Statistic for testing the significance of global effect: + + F ( c ) + ( c ) + ( c )( c ) X X error XX N cc Statistic for testing significance Statistic for testing significance of each factor s main-effect: of interaction-effect: X X X ( c ) ( c )( c ) F F error error N c c N c c

12 Two-Way Analysis of Variance Source of Sum of Mean Sig. of Variation squares df square F F w Main Effects Promotion Coupon Combined Two-way interaction Model Residual (error) TOTAL Cell Means Promotion Coupon Count Mean High es High No Medium es Medium No Low es Low No TOTAL 30 Factor Level Means Promotion Coupon Count Mean High Medium Low es No Grand Mean

13 Store sales Promotion coupon no coupon low 5,4,0 medium 7,6 4,8 high 9, 7,4 Effect of promotion level on sales for alternative coupon levels Sales coupon no coupon 0 low medium high Promotion 5 Patterns of Interaction Case : No Interaction Case : Ordinal Interaction X X es es X X No No X X X 3 Promotion X X X 3 Promotion Case 3: Disordinal Interaction: Noncrossover Case 3: Disordinal Interaction: Crossover es X X es X No No X X X X 3 Promotion X X X 3 Promotion 6 3

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