Chapter 18 - Hypothesis testing: Means and proportions. In this chapter, the following content will be discussed: Testing hypothesis about a single mean, hypothesis testing about a single mean – step by step, relationship between confidence interval and hypothesis testing,...
1 Marketing Research Aaker, Kumar, Leone and Day Twelfth Edition Instructor’s Chapter Eighteen Hypothesis Testing: Means and Proportions Marketing Research 12th Edition Testing Hypothesis about a Single Mean • Broadly speaking, the choice of the distribution depends on (1) (2) (3) the purpose of hypothesis testing, the size of the sample, and whether or not the population standard deviation is known. Marketing Research 12th Edition Hypothesis Testing about a Single Mean – Step by Step Marketing Research 12th Edition Hypothesis Testing About A Single Mean Example 1 Twotailed test • Ho: = 5000 (hypothesized value of population) • Ha: 5000 (alternative hypothesis) • n = 100 • X = 4960 • = 250 • = 0.05 Marketing Research 12th Edition Rejection rule: if |zcalc| > z /2 then reject Ho 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 Marketing Research 12th Edition Rejection rule: if |tcalc| > tdf, /2 then reject Ho 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 Marketing Research 12th Edition Relationship between confidence interval and hypothesis testing • Hypothesis testing and Confidence Intervals are two sides of the same coin t (X ) sx X ts x interval estimate of Marketing Research 12th Edition Analysis of Variance • • ANOVA mainly used for analysis of experimental data Ratio of “betweentreatment” variance and “within treatment” variance • Response variable dependent variable (Y) • Factor (s) independent variables (X) • Treatments different levels of factors (r1, r2, r3, …) Marketing Research 12th Edition 10 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 Marketing Research 12th Edition 11 One Factor Analysis of Variance (Contd.) • • • Betweentreatment variance Variance in the response variable for different treatments Withintreatment 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 Marketing Research 12th Edition One Factor Analysis of Variance – Example Price Level Observations 2 Total Sample mean (Xp) 39 ¢ 12 10 11 50 10 44 ¢ 10 40 49 ¢ 35 12 Overall sample mean: Xp = 8.333 Overall sample size: n = 15 No. of observations per price level,n p=5 Marketing Research 12th Edition 13 Price Experiment ANOVA Table Marketing Research 12th Edition Hypothesis Testing For Differences Between Means • • 14 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 Marketing Research 12th Edition 15 Procedure for Testing of Two Means Marketing Research 12th Edition 16 Hypothesis Testing of Proportions Example • • CEO of a company finds 87% of 225 bulbs to be defect free To Test the hypothesis that 95% of the bulbs are defect free Po = .95: hypothesized value of the proportion of defectfree bulbs qo = .05: hypothesized value of the proportion of defective bulbs p = .87: sample proportion of defectfree bulbs q = .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 Marketing Research 12th Edition 17 Hypothesis Testing of Proportions – Example (Contd.) • Standard error = • Using Zvalue for .95 as 1.96, the limits of the acceptance region are • Therefore, Reject Null hypothesis Marketing Research 12th Edition 18 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 Marketing Research 12th Edition 19 Hypothesis Testing of Difference between Proportions – Example (contd.) Marketing Research 12th Edition 20 The Probability Values (pvalue) Approach to Hypothesis Testing Difference between using and pvalue • • Hypothesis testing with a prespecified ▫ Researcher determines "is the probability of what has been observed less than ?" ▫ Reject or fail to reject ho accordingly Using the pvalue: ▫ ▫ 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 prespecified significance level Marketing Research 12th Edition 21 The Probability Values (pvalue) Approach to Hypothesis Testing (Contd.) • • • • pvalue provides researcher with alternative method of testing hypothesis without prespecifying pvalue is the largest level of significance at which we would not reject ho In general, the smaller the pvalue, the greater the confidence in sample findings pvalue is generally sensitive to sample size ▫ • A large sample should yield a low pvalue pvalue can report the impact of the sample size on the reliability of the results Marketing Research 12th Edition 22 ProbabilityValuesApproachtoHypothesis Testing Example: NullhypothesisH0:à=25 ▫ Alternative hypothesis Ha : µ ≠ 25 Sample size n = 50 Sample mean X =25.2 Standard deviation = 0.7 ▫ ▫ ▫ Standard error = Z statistic = Pvalue = 2 X 0.0228 = 0.0456 (twotailed test) At α = 0.05, reject null hypothesis Marketing Research 12th Edition 23 End of Chapter Eighteen Marketing Research 12th Edition ...2 Chapter Eighteen Hypothesis Testing: Means and Proportions Marketing Research 12th Edition Testing Hypothesis about a Single Mean • Broadly speaking, the choice of the distribution ... No. of observations per price level,n p=5 Marketing Research 12th Edition 13 Price Experiment ANOVA Table Marketing Research 12th Edition Hypothesis Testing For Differences Between Means • • 14 Commonly used in experimental research. .. If the population standard deviation is not known, whether they can be assumed to be equal or not Marketing Research 12th Edition 15 Procedure for Testing of Two Means Marketing Research 12th Edition 16 Hypothesis Testing of Proportions Example