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1 Marketing Research Aaker, Kumar, Day Ninth Edition Instructor’s Presentation Slides 2 Chapter Eighteen Hypothesis Testing: Means and Proportions http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 3 Hypothesis Testing For Differences Between Means • Commonly used in experimental research • Statistical technique used is Analysis of Variance (ANOVA) Hypothesis Testing Criteria Depends on: • Whether the samples are obtained from different or related populations • Whether the population is known or not known • If the population standard deviation is not known, whether they can be assumed to be equal or not http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 4 The Probability Values (p-value) Approach to Hypothesis Testing Difference between using and p-value • Hypothesis testing with a pre-specified ▫ Researcher determines "is the probability of what has been observed less than ?" ▫ Reject or fail to reject ho accordingly • Using the p-value: ▫ Researcher determines "how unlikely is the result that has been observed?" ▫ Decide whether to reject or fail to reject ho without being bound by a pre-specified significance level http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 5 The Probability Values (p-value) Approach to Hypothesis Testing (Contd.) • p-value provides researcher with alternative method of testing hypothesis without pre-specifying • p-value is the largest level of significance at which we would not reject ho • In general, the smaller the p-value, the greater the confidence in sample findings • p-value is generally sensitive to sample size ▫ A large sample should yield a low p-value • p-value can report the impact of the sample size on the reliability of the results http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 6 Hypothesis Testing about a Single Mean – Step by Step Formulate Hypotheses Select appropriate formula Select significance level Calculate z or t statistic Calculate degrees of freedom (for t-test) Obtain critical value from table Make decision regarding the Null-hypothesis http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 7 Hypothesis Testing About A Single Mean Example 1 - Two-tailed test • Ho: = 5000 (hypothesized value of population) • Ha: 5000 (alternative hypothesis) • n = 100 • X = 4960 • = 250 • = 0.05 Rejection rule: if |zcalc| > z/2 then reject Ho http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 8 Hypothesis Testing About A Single Mean Example 2 • Ho: = 1000 (hypothesized value of population) • Ha: 1000 (alternative hypothesis) • n = 12 • X = 1087.1 • s = 191.6 • = 0.01 Rejection rule: if |tcalc| > tdf, /2 then reject Ho http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 9 Hypothesis Testing About A Single Mean Example 3 • Ho: 1000 (hypothesized value of population) • Ha: > 1000 (alternative hypothesis) • n = 12 • X = 1087.1 • s = 191.6 • = 0.05 Rejection rule: if tcalc > tdf, then reject Ho http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 10 Confidence Intervals • Hypothesis testing and Confidence Intervals are two sides of the same coin. (X ) t sx http://www.drvkumar.com/mr10/ X tsx interval estimate of Marketing Research 10th Edition 11 Procedure for Testing of Two Means http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 12 Hypothesis Testing of Proportions - Example • CEO of a company finds 87% of 225 bulbs to be defectfree • To Test the hypothesis that 95% of the bulbs are defect free Po qo p q = .95: hypothesized value of the proportion of defect-free bulbs = .05: hypothesized value of the proportion of defective bulbs = .87: sample proportion of defect-free bulbs = .13: sample proportion of defective bulbs Null hypothesis Ho: p = 0.95 Alternative hypothesis Ha: p ≠ 0.95 Sample size n = 225 Significance level = 0.05 http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 13 Hypothesis Testing of Proportions – Example (Contd.) • Standard error = • Using Z-value for .95 as 1.96, the limits of the acceptance region are • Therefore, Reject Null hypothesis http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 14 Hypothesis Testing of Difference between Proportions - Example • Competition between sales reps, John and Linda for converting prospects to customers: PJ = .84 John’s conversion ratio based on this sample of prospects qJ = .16 Proportion that John failed to convert n1 = 100 John’s prospect sample size pL = .82 Linda’s conversion ratio based on her sample of prospects qL = .18 Proportion that Linda failed to convert n2 = 100 Linda’s prospect sample size Null hypothesis Ho: PJ = P L Alternative hypothesis Ha : PJ ≠ PL Significance level α = .05 http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 15 Hypothesis Testing of Difference between Proportions – Example (contd.) http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 16 Probability –Values Approach to Hypothesis Testing • Example: ▫ Null hypothesis H0 : µ = 25 ▫ ▫ ▫ ▫ Alternative hypothesis Ha : µ ≠ 25 Sample size n = 50 Sample mean X =25.2 Standard deviation = 0.7 Standard error = Z- statistic = P-value = 2 X 0.0228 = 0.0456 (two-tailed test) At α = 0.05, reject null hypothesis http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 17 Analysis of Variance • ANOVA mainly used for analysis of experimental data • Ratio of “between-treatment” variance and “withintreatment” variance • Response variable - dependent variable (Y) • Factor (s) - independent variables (X) • Treatments - different levels of factors (r1, r2, r3, …) http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 18 One - Factor Analysis of Variance • Studies the effect of 'r' treatments on one response variable • Determine whether or not there are any statistically significant differences between the treatment means 1, 2,... R Ho: all treatments have same effect on mean responses H1 : At least 2 of 1, 2 ... r are different http://www.drvkumar.com/mr10/ Marketing Research 10th Edition 19 One - Factor Analysis of Variance (Contd.) • Between-treatment variance - Variance in the response variable for different treatments. • Within-treatment variance - Variance in the response variable for a given treatment. • If we can show that ‘‘between’’ variance is significantly larger than the ‘‘within’’ variance, then we can reject the null hypothesis http://www.drvkumar.com/mr10/ Marketing Research 10th Edition One - Factor Analysis of Variance – Example Price Level Observations Sample mean (Xp) 1 2 2 4 5 Total 39 ¢ 8 12 10 9 11 50 10 44 ¢ 7 10 6 8 9 40 8 49 ¢ 4 8 7 9 7 35 7 http://www.drvkumar.com/mr10/ Overall sample mean: Xp = 8.333 Overall sample size: n = 15 No. of observations per price level,n p=5 20 Marketing Research 10th Edition 21 Price Experiment ANOVA Table http://www.drvkumar.com/mr10/ Marketing Research 10th Edition