1. PowerPoint Presentation by Charlie Cook The University of West Alabama William G. Zikmund Barry J. Babin 9 th Edition Part 5 Sampling and Fieldwork © 2007 Thomson/South-Western. All rights reserved. Chapter 17 Determination of Sample Size: A Review of Statistical Theory

2. © 2007 Thomson/South-Western. All rights reserved.17–2 LEARNING OUTCOMES 1.Implement descriptive and inferential statistics 2.Interpret frequency distributions, proportions, and measures of central tendency and dispersion 3.Distinguish among population, sample, and sampling distributions 4.Explain the central-limit theorem 5.Summarize the use of confidence interval estimates 6.Discuss major issues in specifying sample size After studying this chapter, you should be able to

3. © 2007 Thomson/South-Western. All rights reserved.17–3 Review of Basic Terminology Descriptive StatisticsDescriptive Statistics  Describe characteristics of populations or samples. Inferential StatisticsInferential Statistics  Make inferences about whole populations from a sample. Sample StatisticsSample Statistics  Variables in a sample or measures computed from sample data. Population ParametersPopulation Parameters  Variables in a population or measured characteristics of the population.

4. © 2007 Thomson/South-Western. All rights reserved.17–4 What does Statistics Mean? Descriptive statisticsDescriptive statistics  Number of people  Trends in employment  Data Inferential statisticsInferential statistics  Make an inference about a population from a sample.

5. © 2007 Thomson/South-Western. All rights reserved.17–5 Making Data Usable Frequency DistributionFrequency Distribution  A set of data organized by summarizing the number of times a particular value of a variable occurs. Percentage DistributionPercentage Distribution  A frequency distribution organized into a table (or graph) that summarizes percentage values associated with particular values of a variable. ProbabilityProbability  The long-run relative frequency with which an event will occur.

6. © 2007 Thomson/South-Western. All rights reserved.17–6 EXHIBIT 17.1 Frequency Distribution of Deposits

7. © 2007 Thomson/South-Western. All rights reserved.17–7 EXHIBIT 17.2 Percentage Distribution of Deposits

8. © 2007 Thomson/South-Western. All rights reserved.17–8 EXHIBIT 17.3 Probability Distribution of Deposits

9. © 2007 Thomson/South-Western. All rights reserved.17–9 Making Data Usable (cont’d) ProportionProportion  The percentage of elements that meet some criterion Measures of Central TendencyMeasures of Central Tendency  Mean: the arithmetic average.  Median: the midpoint; the value below which half the values in a distribution fall.  Mode: the value that occurs most often. Population MeanSample Mean

10. © 2007 Thomson/South-Western. All rights reserved.17–10 EXHIBIT 17.4 Number of Sales Calls per Day by Salesperson

11. © 2007 Thomson/South-Western. All rights reserved.17–11 Measures of Dispersion The RangeThe Range  The distance between the smallest and the largest values of a frequency distribution Deviation ScoresDeviation Scores  Indicate how far any observation is from the mean  Example:

12. © 2007 Thomson/South-Western. All rights reserved.17–12 EXHIBIT 17.6 Low Dispersion versus High Dispersion

13. © 2007 Thomson/South-Western. All rights reserved.17–13 EXHIBIT 17.5 Sales Levels for Two Products with Identical Average Sales

14. © 2007 Thomson/South-Western. All rights reserved.17–14 Measures of Dispersion (cont’d) Why Use the Standard Deviation?Why Use the Standard Deviation?  Variance  A measure of variability or dispersion. Its square root is the standard deviation.  Standard deviation  A quantitative index of a distribution’s spread, or variability; the square root of the variance for a distribution.  The average of the amount of variance for a distribution.  Used to calculate the likelihood (probability) of an event occurring.

15. © 2007 Thomson/South-Western. All rights reserved.17–15 Calculating Deviation Standard Deviation =

16. © 2007 Thomson/South-Western. All rights reserved.17–16 EXHIBIT 17.7 Calculating a Standard Deviation: Number of Sales Calls per Day for Eight Salespeople a The summation of this column is not used in the calculation of the standard deviation.

17. © 2007 Thomson/South-Western. All rights reserved.17–17 The Normal Distribution Normal DistributionNormal Distribution  A symmetrical, bell-shaped distribution (normal curve) that describes the expected probability distribution of many chance occurrences.  99% of its values are within ± 3 standard deviations from its mean.  Example: IQ scores Standardized Normal DistributionStandardized Normal Distribution  A purely theoretical probability distribution that reflects a specific normal curve for the standardized value, z.

18. © 2007 Thomson/South-Western. All rights reserved.17–18 EXHIBIT 17.8 Normal Distribution: Distribution of Intelligence Quotient (IQ) Scores

19. © 2007 Thomson/South-Western. All rights reserved.17–19 The Normal Distribution (cont’d) Characteristics of a Standardized Normal DistributionCharacteristics of a Standardized Normal Distribution 1. It is symmetrical about its mean. 2. The mean identifies the normal curve’s highest point (the mode) and the vertical line about which this normal curve is symmetrical. 3. The normal curve has an infinite number of cases (it is a continuous distribution), and the area under the curve has a probability density equal to 1.0. 4. The standardized normal distribution has a mean of 0 and a standard deviation of 1.

20. © 2007 Thomson/South-Western. All rights reserved.17–20 EXHIBIT 17.9 Standardized Normal Distribution

21. © 2007 Thomson/South-Western. All rights reserved.17–21 The Normal Distribution (cont’d) Standardized ValuesStandardized Values  Used to compare an individual value to the population mean in units of the standard deviation

22. © 2007 Thomson/South-Western. All rights reserved.17–22 EXHIBIT 17.10 Standardized Normal Table: Area under Half of the Normal Curve a a Area under the segment of the normal curve extending (in one direction) from the mean to the point indicated by each row-column combination. For example, about 68 percent of normally distributed events can be expected to fall within 1.0 standard deviation on either side of the mean (0.341 × 2). An interval of almost 2.0 standard deviations around the mean will include 95 percent of all cases.

23. © 2007 Thomson/South-Western. All rights reserved.17–23 EXHIBIT 17.11 Linear Transformation of Any Normal Variable into a Standardized Normal Variable Source: Thomas H. Wonnacott and Ronald J. Wonnacott, Introductory Statistics (New York: John Wiley & Sons, 1969), p. 70. Reprinted with permission.

24. © 2007 Thomson/South-Western. All rights reserved.17–24 EXHIBIT 17.12 Standardized Distribution Curve

25. © 2007 Thomson/South-Western. All rights reserved.17–25 Population Distribution, Sample Distribution, and Sampling Distribution Population DistributionPopulation Distribution  A frequency distribution of the elements of a population. Sample DistributionSample Distribution  A frequency distribution of a sample. Sampling DistributionSampling Distribution  A theoretical probability distribution of sample means for all possible samples of a certain size drawn from a particular population. Standard Error of the MeanStandard Error of the Mean  The standard deviation of the sampling distribution.

26. © 2007 Thomson/South-Western. All rights reserved.17–26 Three Important Distributions

27. © 2007 Thomson/South-Western. All rights reserved.17–27 EXHIBIT 17.13 Fundamental Types of Distributions Source: Adapted from D. H. Sanders, A. F. Murphy, and R. J. Eng, Statistics: A Fresh Approach (New York: McGraw-Hill, 1980), p. 123.

28. © 2007 Thomson/South-Western. All rights reserved.17–28 Central-limit Theorem Central-limit TheoremCentral-limit Theorem  The theory that, as sample size increases, the distribution of sample means of size n, randomly selected, approaches a normal distribution.

29. © 2007 Thomson/South-Western. All rights reserved.17–29 EXHIBIT 17.14 Distribution of Sample Means for Samples of Various Sizes and Population Distributions

30. © 2007 Thomson/South-Western. All rights reserved.17–30 EXHIBIT 17.15 Population Distribution: Hypothetical Toy Expenditures

31. © 2007 Thomson/South-Western. All rights reserved.17–31 EXHIBIT 17.16 Calculation of Population Mean

32. © 2007 Thomson/South-Western. All rights reserved.17–32 EXHIBIT 17.17 Arithmetic Means of Samples and Frequency Distribution of Sample Means

33. © 2007 Thomson/South-Western. All rights reserved.17–33 Estimation of Parameters Point EstimatesPoint Estimates  An estimate of the population mean in the form of a single value, usually the sample mean.  Gives no information about the possible magnitude of random sampling error. Confidence Interval EstimateConfidence Interval Estimate  A specified range of numbers within which a population mean is expected to lie.  An estimate of the population mean based on the knowledge that it will be equal to the sample mean plus or minus a small sampling error.

34. © 2007 Thomson/South-Western. All rights reserved.17–34 Estimation of Parameters Confidence LevelConfidence Level  A percentage or decimal value that tells how confident a researcher can be about being correct.  It states the long-run percentage of confidence intervals that will include the true population mean.  The crux of the problem for a researcher is to determine how much random sampling error to tolerate.  Traditionally, researchers have used the 95% confidence level (a 5% tolerance for error).

35. © 2007 Thomson/South-Western. All rights reserved.17–35 Calculating a Confidence Interval Estimation of the sampling error Approximate location (value) of the population mean

36. © 2007 Thomson/South-Western. All rights reserved.17–36 Calculating a Confidence Interval (cont’d)

37. © 2007 Thomson/South-Western. All rights reserved.17–37 Sample Size Random Error and Sample SizeRandom Error and Sample Size  Random sampling error varies with samples of different sizes.  Increases in sample size reduce sampling error at a decreasing rate.  Diminishing returns: random sampling error is inversely proportional to the square root of n.

38. © 2007 Thomson/South-Western. All rights reserved.17–38 EXHIBIT 17.19 Statistical Information Needed to Determine Sample Size for Questions Involving Means

39. © 2007 Thomson/South-Western. All rights reserved.17–39 Factors of Concern in Choosing Sample Size Variance (or Heterogeneity)Variance (or Heterogeneity)  A heterogeneous population has more variance (a larger standard deviation) which will require a larger sample.  A homogeneous population has less variance (a smaller standard deviation) which permits a smaller sample. Magnitude of Error (Confidence Interval)Magnitude of Error (Confidence Interval)  How precise must the estimate be? Confidence LevelConfidence Level  How much error will be tolerated?

40. © 2007 Thomson/South-Western. All rights reserved.17–40 EXHIBIT 17.18 Relationship between Sample Size and Error Source: From Foundations of Behavioral Research 3rd edition by Kerlinger. © 1986. Reprinted with permission of Wadsworth, a division of Thomson Learning: www.thomsonrights.com. Fax 800-730-2215.

41. © 2007 Thomson/South-Western. All rights reserved.17–41 Estimating Sample Size for Questions Involving Means Sequential SamplingSequential Sampling  Conducting a pilot study to estimate the population parameters so that another, larger sample of the appropriate sample size may be drawn. Estimating sample size:Estimating sample size:

42. © 2007 Thomson/South-Western. All rights reserved.17–42 Sample Size Example Suppose a survey researcher, studying expenditures on lipstick, wishes to have a 95 percent confident level ( Z ) and a range of error ( E ) of less than $2.00. The estimate of the standard deviation is $29.00. What is the calculated sample size?Suppose a survey researcher, studying expenditures on lipstick, wishes to have a 95 percent confident level ( Z ) and a range of error ( E ) of less than $2.00. The estimate of the standard deviation is $29.00. What is the calculated sample size?

43. © 2007 Thomson/South-Western. All rights reserved.17–43 Sample Size Example Suppose, in the same example as the one before, the range of error ( E ) is acceptable at $4.00. Sample size is reduced.Suppose, in the same example as the one before, the range of error ( E ) is acceptable at $4.00. Sample size is reduced.

44. © 2007 Thomson/South-Western. All rights reserved.17–44 Calculating Sample Size at the 99 Percent Confidence Level

45. © 2007 Thomson/South-Western. All rights reserved.17–45 Determining Sample Size for Proportions

46. © 2007 Thomson/South-Western. All rights reserved.17–46 Determining Sample Size for Proportions (cont’d)

47. © 2007 Thomson/South-Western. All rights reserved.17–47 753  001225. 922.  001225 )24)(.8416.3(  )035(. )4 )(. 6(.) 96 1. ( n 4.q 6.p 2 2    Calculating Example Sample Size at the 95 Percent Confidence Level

48. © 2007 Thomson/South-Western. All rights reserved.17–48 EXHIBIT 17.20 Selected Tables for Determining Sample Size When the Characteristic of Interest Is a Proportion a In these cases, more than 50 percent of the population is required in the sample. Since the normal approximation of the hypergeometric distribution is a poor approximation in such instances, no sample value is given. Source: Nan Lin, Foundations of Social Research (New York: McGraw-Hill, 1976), p. 447. Copyright © 1976 by Nan Lin. Used with permission.

49. © 2007 Thomson/South-Western. All rights reserved.17–49 EXHIBIT 17.20 Selected Tables for Determining Sample Size When the Characteristic of Interest Is a Proportion (cont’d) a In these cases, more than 50 percent of the population is required in the sample. Since the normal approximation of the hypergeometric distribution is a poor approximation in such instances, no sample value is given. Source: Nan Lin, Foundations of Social Research (New York: McGraw-Hill, 1976), p. 447. Copyright © 1976 by Nan Lin. Used with permission.

50. © 2007 Thomson/South-Western. All rights reserved.17–50 EXHIBIT 17.21 Allowance for Random Sampling Error (Plus and Minus Percentage Points) at 95 Percent Confidence Level

51. © 2007 Thomson/South-Western. All rights reserved.17–51 Key Terms and Concepts sample statisticssample statistics population parameterspopulation parameters frequency distributionfrequency distribution percentage distributionpercentage distribution probabilityprobability proportionproportion meanmean medianmedian modemode variancevariance standard deviationstandard deviation normal distributionnormal distribution standardized normal distributionstandardized normal distribution population distributionpopulation distribution sample distributionsample distribution sampling distributionsampling distribution standard error of the meanstandard error of the mean central-limit theoremcentral-limit theorem point estimatepoint estimate confidence interval estimateconfidence interval estimate confidence levelconfidence level