Hypothesis Testing: Means and Proportions

Marketing ResearchAaker, Kumar, Leone and Day Twelfth EditionInstructor’s Presentation Slides1Chapter Eighteen2Hypothesis Testing: Means and ProportionsMarketing Research 12th Edition Testing Hypothesis about a Single MeanBroadly speaking, the choice of the distribution depends on the purpose of hypothesis testing, the size of the sample, andwhether or not the population standard deviation is known. 3Population standarddeviation σ is knownSample size does not really matterPopulation standarddeviation σ is NOT knownThe size of the sample dictates the choice of the probability standard distributionMarketing Research 12th Edition Hypothesis Testing about a Single Mean – Step by StepFormulate HypothesesSelect appropriate formulaSelect significance levelCalculate z or t statisticCalculate degrees of freedom (for t-test)Obtain critical value from tableMake decision regarding the Null-hypothesis4Marketing Research 12th Edition Hypothesis Testing About A Single Mean - Example 1 - Two-tailed testHo:  = 5000 (hypothesized value of population) Ha:   5000 (alternative hypothesis)n = 100X = 4960  = 250 = 0.05 Rejection rule: if |zcalc| > z/2 then reject Ho5Marketing Research 12th Edition Hypothesis Testing About A Single Mean - Example 2Ho:  = 1000 (hypothesized value of population) Ha:   1000 (alternative hypothesis)n = 12X = 1087.1s = 191.6 = 0.01 Rejection rule: if |tcalc| > tdf, /2 then reject Ho6Marketing Research 12th Edition Hypothesis Testing About A Single Mean - Example 3Ho:   1000 (hypothesized value of population) Ha:  > 1000 (alternative hypothesis)n = 12X = 1087.1s = 191.6 = 0.05Rejection rule: if tcalc > tdf,  then reject Ho7Marketing Research 12th Edition Relationship between confidenceinterval and hypothesis testingHypothesis testing and Confidence Intervals are two sides of the same coin. 8Marketing Research 12th Edition Analysis of VarianceANOVA mainly used for analysis of experimental dataRatio of “between-treatment” variance and “within- treatment” varianceResponse variable - dependent variable (Y)Factor (s) - independent variables (X)Treatments - different levels of factors (r1, r2, r3, )9Marketing Research 12th Edition One - Factor Analysis of VarianceStudies the effect of 'r' treatments on one response variableDetermine whether or not there are any statistically significant differences between the treatment means 1, 2,... R10Ho: all treatments have same effect on mean responses H1 : At least 2 of 1, 2 ... r are differentMarketing Research 12th Edition 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 hypothesis11Marketing Research 12th Edition One - Factor Analysis of Variance – ExampleObservationsSample mean (Xp) 12245Total39 ¢81210911501044 ¢71068940849 ¢4879735712Price LevelOverall sample mean: Xp = 8.333Overall sample size: n = 15No. of observations per price level,n p=5Marketing Research 12th Edition Price Experiment ANOVA Table13Marketing Research 12th Edition Hypothesis Testing For Differences Between MeansCommonly used in experimental researchStatistical technique used is Analysis of Variance (ANOVA)14Hypothesis 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 notMarketing Research 12th Edition Procedure for Testing of Two Means15Marketing Research 12th Edition Hypothesis Testing of Proportions - ExampleCEO of a company finds 87% of 225 bulbs to be defect-freeTo Test the hypothesis that 95% of the bulbs are defect free16Po = .95: hypothesized value of the proportion of defect-free bulbsqo = .05: hypothesized value of the proportion of defective bulbsp = .87: sample proportion of defect-free bulbsq = .13: sample proportion of defective bulbsNull hypothesis Ho: p = 0.95Alternative hypothesis Ha: p ≠ 0.95Sample size n = 225Significance level = 0.05Marketing Research 12th Edition Hypothesis Testing of Proportions – Example (Contd.)Standard error = Using Z-value for .95 as 1.96, the limits of the acceptance region areTherefore, Reject Null hypothesis17Marketing Research 12th Edition Hypothesis Testing of Difference between Proportions - ExampleCompetition between sales reps, John and Linda for converting prospects to customers:18Null hypothesis Ho: PJ = P LAlternative hypothesis Ha : PJ ≠ PL Significance level α = .05PJ = .84 John’s conversion ratio based on this sample of prospectsqJ = .16 Proportion that John failed to convertn1 = 100 John’s prospect sample sizepL = .82 Linda’s conversion ratio based on her sample of prospectsqL = .18 Proportion that Linda failed to convertn2 = 100 Linda’s prospect sample sizeMarketing Research 12th Edition Hypothesis Testing of Difference between Proportions – Example (contd.)19Marketing Research 12th Edition The Probability Values (p-value) Approach to Hypothesis TestingDifference between using  and p-valueHypothesis testing with a pre-specified Researcher determines "is the probability of what has been observed less than ?"Reject or fail to reject ho accordinglyUsing 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 level20Marketing Research 12th Edition 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 hoIn general, the smaller the p-value, the greater the confidence in sample findingsp-value is generally sensitive to sample sizeA large sample should yield a low p-valuep-value can report the impact of the sample size on the reliability of the results21Marketing Research 12th Edition Probability –Values Approach to Hypothesis TestingExample:Null hypothesis H0 : µ = 25Alternative hypothesis Ha : µ ≠ 25Sample size n = 50Sample mean X =25.2Standard deviation = 0.7Standard error = Z- statistic = P-value = 2 X 0.0228 = 0.0456 (two-tailed test) At α = 0.05, reject null hypothesis22Marketing Research 12th Edition 23End of Chapter EighteenMarketing Research 12th Edition