Transcript Chapter 13
• • • • • • • • • OPENING QUESTIONS 1. What key concepts and symbols are pertinent to sampling? 2. How are the sampling distribution, statistical inference, and standard error relevant to sampling? 3. What is the statistical approach to determining sample size based on simple random sampling and the construction of confidence intervals? 4. How can we derive the formulas to statistically determine the sample size for estimating means and proportions? 5. How should the sample size be adjusted to account for incidence and completion rates? 6. Why is it difficult to statistically determine the sample size in international marketing research? 7. What is the interface of technology with sample size determination? 8. What ethical issues are related to sample size determination, particularly the estimation of population variance? Figure 13.1 Relationship of Sample Size Determination to the Previous Chapters and the Marketing Research Process Figure 13.1 Relationship to the Previous Chapters & The Marketing Research Process Focus of This Chapter Relationship to Previous Chapters • Statistical Approach to Determining Sample Size • Research Design Components (Chapter 3) • Adjusting the Statistically Determined Sample Size • Sampling Design Process (Chapter 12) Relationship to Marketing Research Process Problem Definition Approach to Problem Research Design Field Work Data Preparation and Analysis Report Preparation and Presentation Figure 13.2 Final and Initial Sample Size Determination: An Overview Be an MR! Be a DM! Definitions and Symbols Table 13.1 The Sampling Distribution Figs 13a.113a.3 Appendix 13a Statistical Approach to Determining Sample Size Fig 13.3 Confidence Interval Approach Table 13.2 Fig 13.4 Adjusting the Statistically Determined Sample Size Application to Contemporary Issues International Technology Ethics What Would You Do? Experiential Learning Opening Vignette Definitions and Symbols • Parameter: A parameter is a summary description of a fixed characteristic or measure of the target population. A parameter denotes the true value which would be obtained if a census rather than a sample was undertaken. • Statistic: A statistic is a summary description of a characteristic or measure of the sample. The sample statistic is used as an estimate of the population parameter. • Sampling Distribution: A distribution of the values of a sample statistic, for example, the sample mean. Definitions and Symbols • Central Limit Theorem: as the sample size increases, the distribution of the sample mean of a randomly selected sample approaches normal • Precision level: When estimating a population parameter by using a sample statistic, the precision level is the desired size of the estimating interval. This is the maximum permissible difference between the sample statistic and the population parameter. • Confidence interval: The confidence interval is the range into which the true population parameter will fall, assuming a given level of confidence. • Confidence level: The confidence level is the probability that a confidence interval will include the population parameter. TABLE 13.1 Symbols for Population and Sample Variables ____________________________________________________________ Variable Population Sample ____________________________________________________________ Mean X Proportion p Variance s Standard deviation s Size N n Standard error of the mean x Sx proportion p Standardized variate (z) X – Sp X –X Sx 2 2 Standard error of the ___________________________________________________________ Figure 13.3 The Confidence Interval Approach and Determining Sample Size Confidence Interval Approach Means Proportions The Confidence Interval Approach Calculation of the confidence interval involves determining a distance below (X L) and above (X U) the population mean ( ), which contains a specified area of the normal curve. The z values corresponding to XL and XU are X - L zL = x zU = XU - x The Confidence Interval Approach where zL = –z and zU = +z. Therefore, the X L = - zx X U = + zx The Confidence Interval Approach The confidence interval is given by X zx We can now set a 95% confidence interval around the sample mean of $182. x = n = 55/ 300 = 3.18 The 95% confidence interval is given by X + 1.96x = 182.00 + 1.96(3.18) = 182.00 + 6.23 Thus the 95% confidence interval ranges from $175.77 to $188.23. Figure 13.4 95% Confidence Interval 0.475 _ XL 0.475 _ X _ XU Figure 13A.1 Finding Probabilities Corresponding to Known Values Area is 0.3413 Area between µ and µ +1= 0.3431 Area between µ and µ +2= 0.4772 Area between µ and µ +3= 0.4986 µ-3 X Scale (µ=50, =5) 35 Z Scale -3 µ-2 µ-1 µ µ+1 µ+2 µ+3 40 45 50 55 60 65 -2 -1 0 +1 +2 +3 Figure 13A.2 Finding Values Corresponding to Known Probabilities Area is 0.500 Area is 0.450 Area is 0.050 X Scale X 50 Z Scale -Z 0 Figure 13A.3 Finding Values Corresponding to Known Probabilities: Confidence Interval Area is 0.475 Area is 0.475 Area is 0.025 Area is 0.025 X -Z 50 X Scale 0 Z Scal e +Z