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Chapter 17 Sample Size Determination *Adapted from Andrei Strijnev 1 Basic Considerations in Determining Sample Size Precision – the degree of error in a study, or the size of the estimating interval How precise should the estimate be? More – less error, smaller estimation interval Less – more error, greater estimation interval Confidence – the degree to which one can feel confident that an estimate approximates the true value These two terms are highly correlated 2 The Importance of Sample Size Too small sample size leads to undesirably large sampling error. Too large sample size wastes time and money. 3 Methods of Determining Sample Size Ad Hoc Methods Based on budget constraint Use of expert guidance Use previous sample sizes Statistical Method 4 Sample Size Determination: The Intuition More accurate information required => larger sample size needed More diversified population => larger sample size needed The marginal gain in information accuracy is diminishing with the increase of sample size 5 Sample Size Determination: The Equation The relationship between sampling error and sample size: s sx = n sx = sampling error for variable x s = estimated standard deviation for variable x in the population n = sample size 6 Sampling Error & Sample Size 3.5 3 2.5 Sampling 2 Error 1.5 1 0.5 0 0 500 1000 1500 2000 Sample Size 7 Sampling Error & Confidence Interval Marketing researchers usually use confidence intervals to describe sampling error. Meaning of confidence interval If the estimate of a variable is 1000 and it has a 95% confidence interval of 100, then… with 95% probability the true value of that variable is between 900 and 1100. 8 Sample Size Determination: Measures of Interest in Most Surveys Sample Mean, Examples include: Mean rating on an attitudes measurement scale Mean income, age Mean number of purchases, monthly expenditure Sample Proportion, Examples include: Proportion of households owning 2 cars Proportion of households spending more than $400/month on food Probability of purchase 9 The Confidence Interval Approach to Sample Size: Estimates of Means If a 95% confidence interval of I95% is desired, 2 (1.96s) n= 2 ( I 95% ) . 10 Example A marketing manager of a frozen foods firm wants to estimate the average dollars that families in a certain city spend on frozen foods per year. He wants the estimate to have a 95% confidence interval of $10. He estimates that the standard deviation of annual family expenditures is about $100. What is the desired sample size for this study? 11 In this problem, I95%=10, s=100. Applying 2 (1.96s) n= 2 ( I 95% ) , we obtain that n=(1.96*100)2/102=385. Thus, 385 families must be chosen for this study. 12 How to Estimate Standard Deviation (s) 1.Subjective judgment 2.Use pilot study 3.If min and max values in population is known, choose s = (max value - min value) / 6. 13 The Confidence Interval Approach to Sample Size: Estimates of Proportions If a 95% confidence interval of I95% is desired, 2 (1.96) n= p (1 p ) 2 ( I 95% ) , where p is the estimated proportion in the population. 14 Example A sporting goods marketer wants to estimate the proportion of tennis players among college students. He wants the estimate to have a 95% confidence interval of 0.02. A pilot study of 50 students showed that 20 of them played tennis. What is the required sample size for the final study? 15 In this problem, I95%=0.02, p=20/50=0.4. Applying 2 (1.96) n= p (1 p ) , 2 ( I 95% ) we obtain that n=[(1.962)/0.022]*0.4*(1-0.4)=2305. Thus, 2305 students must be sampled. 16 How to Estimate p 1.Subjective judgment 2.Use pilot study 3.Set p=50% 17 More Examples PETsMART wants to determine the average number of cats in each U.S. household. It wants the estimate to have a 95% confidence interval of 0.5 cat. You recall once reading an article that says the households with the most cats was headed by an 85 year old woman who owned 35 cats. What is the desired sample size for this study? 18 More Examples PETsMART wants to determine the proportion of U.S. households who have dogs. It wants its estimate to have a 95% confidence interval of 0.04. What is the desired sample size for this study? 19 Punch-Lines The intuition of sample size determination Determine sample size for estimating means and proportions Three ways of estimating s and p 20