1. Business Research Methods William G. Zikmund Chapter 17: Determination of Sample Size

2. Copyright © 2000 by Harcourt, Inc. All rights reserved. Requests for permission to make copies of any part of the work should be mailed to the following address: Permissions Department, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida 32887-6777.

3. Copyright © 2000 by Harcourt, Inc. All rights reserved. WHAT DOES STATISTICS MEAN? DESCRIPTIVE STATISTICS –NUMBER OF PEOPLE –TRENDS IN EMPLOYMENT –DATA INFERENTIAL STATISTICS –MAKE AN INFERENCE ABOUT A POPULATION FROM A SAMPLE

4. Copyright © 2000 by Harcourt, Inc. All rights reserved. POPULATION PARAMATER VARIABLES IN A POPULATION MEASURED CHARACTERISTICS OF A POPULATION GREEK LOWER-CASE LETTERS AS NOTATION

5. Copyright © 2000 by Harcourt, Inc. All rights reserved. SAMPLE STATISTICS VARIABLES IN A SAMPLE MEASURES COMPUTED FROM SAMPLE DATA ENGLISH LETTERS FOR NOTATION

6. Copyright © 2000 by Harcourt, Inc. All rights reserved. MAKING DATA USABLE FREQUENCY DISTRIBUTIONS PROPORTIONS CENTRAL TENDENCY –MEAN –MEDIAN –MODE MEASURES OF DISPERSION

7. Copyright © 2000 by Harcourt, Inc. All rights reserved. Frequency Distribution of Deposits Frequency (number of people making deposits Amount in each range) less than $3,000 499 $3,000 - $4,999 530 $5,000 - $9,999 562 $10,000 - $14,999 718 $15,000 or more 811 3,120

8. Copyright © 2000 by Harcourt, Inc. All rights reserved. Amount Percent less than $3,000 16 $3,000 - $4,999 17 $5,000 - $9,999 18 $10,000 - $14,999 23 $15,000 or more 26 100 Percentage Distribution of Amounts of Deposits

9. Copyright © 2000 by Harcourt, Inc. All rights reserved. Amount Probability less than $3,000.16 $3,000 - $4,999.17 $5,000 - $9,999.18 $10,000 - $14,999.23 $15,000 or more.26 1.00 Probability Distribution of Amounts of Deposits

10. Copyright © 2000 by Harcourt, Inc. All rights reserved. MEASURES OF CENTRAL TENDENCY MEAN - ARITHMETIC AVERAGE –µ, population;, sample MEDIAN - MIDPOINT OF THE DISTRIBUTION MODE - THE VALUE THAT OCCURS MOST OFTEN

11. Copyright © 2000 by Harcourt, Inc. All rights reserved. POPULATION MEAN

12. Copyright © 2000 by Harcourt, Inc. All rights reserved. SAMPLE MEAN

13. Copyright © 2000 by Harcourt, Inc. All rights reserved. Number of Sales Calls Per Day by Salespersons Number of Salesperson Sales calls Mike 4 Patty 3 Billie 2 Bob 5 John 3 Frank 3 Chuck 1 Samantha 5 26

14. Copyright © 2000 by Harcourt, Inc. All rights reserved. Sales for Products A and B, Both Average 200 Product AProduct B 196150 198160 199176 199181 200192200 200201 201202 201213 201224 202240 202261

15. Copyright © 2000 by Harcourt, Inc. All rights reserved. MEASURES OF DISPERSION THE RANGE STANDARD DEVIATION

16. Copyright © 2000 by Harcourt, Inc. All rights reserved. Measures of Dispersion or Spread Range Mean absolute deviation Variance Standard deviation

17. Copyright © 2000 by Harcourt, Inc. All rights reserved. THE RANGE AS A MEASURE OF SPREAD The range is the distance between the smallest and the largest value in the set. Range = largest value – smallest value

18. Copyright © 2000 by Harcourt, Inc. All rights reserved. DEVIATION SCORES the differences between each observation value and the mean:

19. Copyright © 2000 by Harcourt, Inc. All rights reserved. Low Dispersion Verses High Dispersion 150 160 170 180 190 200210 5432154321 Low Dispersion Value on Variable Frequency

20. Copyright © 2000 by Harcourt, Inc. All rights reserved. 150 160 170 180 190 200210 5432154321 Frequency High dispersion Value on Variable

21. Copyright © 2000 by Harcourt, Inc. All rights reserved. AVERAGE DEVIATION

22. Copyright © 2000 by Harcourt, Inc. All rights reserved. MEAN SQUARED DEVIATION

23. Copyright © 2000 by Harcourt, Inc. All rights reserved. THE VARIANCE

24. Copyright © 2000 by Harcourt, Inc. All rights reserved. VARIANCE

25. Copyright © 2000 by Harcourt, Inc. All rights reserved. The variance is given in squared units The standard deviation is the square root of variance:

26. Copyright © 2000 by Harcourt, Inc. All rights reserved. SAMPLE STANDARD DEVIATION

27. Copyright © 2000 by Harcourt, Inc. All rights reserved. POPULATION STANDARD DEVIATION

28. Copyright © 2000 by Harcourt, Inc. All rights reserved. SAMPLE STANDARD DEVIATION

29. Copyright © 2000 by Harcourt, Inc. All rights reserved. SAMPLE STANDARD DEVIATION

30. Copyright © 2000 by Harcourt, Inc. All rights reserved. THE NORMAL DISTRIBUTION NORMAL CURVE BELL-SHAPPED ALMOST ALL OF ITS VALUES ARE WITHIN PLUS OR MINUS 3 STANDARD DEVIATIONS I.Q. IS AN EXAMPLE

31. Copyright © 2000 by Harcourt, Inc. All rights reserved. NORMAL DISTRIBUTION MEAN

32. Copyright © 2000 by Harcourt, Inc. All rights reserved. 2.14% 13.59% 34.13% 13.59% Normal Distribution 2.14%

33. Copyright © 2000 by Harcourt, Inc. All rights reserved. Normal Curve: IQ Example 85115 100 14570

34. Copyright © 2000 by Harcourt, Inc. All rights reserved. STANDARDIZED NORMAL DISTRIBUTION SYMETRICAL ABOUT ITS MEAN MEAN IDENFITIES HIGHEST POINT INFINITE NUMBER OF CASES - A CONTINUOUS DISTRIBUTION AREA UNDER CURVE HAS A PROBABLITY DENSITY = 1.0 MEAN OF ZERO, STANDARD DEVIATION OF 1

35. Copyright © 2000 by Harcourt, Inc. All rights reserved. STANDARD NORMAL CURVE The curve is bell-shaped or symmetrical about 68% of the observations will fall within 1 standard deviation of the mean, about 95% of the observations will fall within approximately 2 (1.96) standard deviations of the mean, almost all of the observations will fall within 3 standard deviations of the mean.

36. Copyright © 2000 by Harcourt, Inc. All rights reserved. A STANDARDIZED NORMAL CURVE 0 1 -2 2 z

37. Copyright © 2000 by Harcourt, Inc. All rights reserved. The Standardized Normal is the Distribution of Z –z+z

38. Copyright © 2000 by Harcourt, Inc. All rights reserved. STANDARDIZED SCORES

39. Copyright © 2000 by Harcourt, Inc. All rights reserved. Standardized Values Used to compare an individual value to the population mean in units of the standard deviation

40. Copyright © 2000 by Harcourt, Inc. All rights reserved. Linear Transformation of Any Normal Variable into a Standardized Normal Variable -2 -1 0 1 2 Sometimes the scale is stretched Sometimes the scale is shrunk    X

41. Copyright © 2000 by Harcourt, Inc. All rights reserved. Population Distribution Sample Distribution Sampling Distribution

42. Copyright © 2000 by Harcourt, Inc. All rights reserved. POPULATION DISTRIBUTION  x 

43. Copyright © 2000 by Harcourt, Inc. All rights reserved. SAMPLE DISTRIBUTION  X S

44. Copyright © 2000 by Harcourt, Inc. All rights reserved. SAMPLING DISTRIBUTION

45. Copyright © 2000 by Harcourt, Inc. All rights reserved. STANDARD ERROR OF THE MEAN STANDARD DEVIATION OF THE SAMPLING DISTRIBUTION

46. Copyright © 2000 by Harcourt, Inc. All rights reserved. STANDARD ERROR OF THE MEAN

47. Copyright © 2000 by Harcourt, Inc. All rights reserved.

48. PARAMETER ESTIMATES POINT ESTIMATES CONFIDENCE INTERVAL ESTIMATES

49. Copyright © 2000 by Harcourt, Inc. All rights reserved. CONFIDENCE INTERVAL

50. Copyright © 2000 by Harcourt, Inc. All rights reserved.

53. ESTIMATING THE STANDARD ERROR OF THE MEAN

54. Copyright © 2000 by Harcourt, Inc. All rights reserved.

55. RANDOM SAMPLING ERROR AND SAMPLE SIZE ARE RELATED

56. Copyright © 2000 by Harcourt, Inc. All rights reserved. SAMPLE SIZE VARIANCE (STANDARD DEVIATION) MAGNITUDE OF ERROR CONFIDENCE LEVEL

57. Copyright © 2000 by Harcourt, Inc. All rights reserved. Sample Size Formula

58. Copyright © 2000 by Harcourt, Inc. All rights reserved. Sample Size Formula - 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.

59. Copyright © 2000 by Harcourt, Inc. All rights reserved. Sample Size Formula - example

60. Copyright © 2000 by Harcourt, Inc. All rights reserved. Sample Size Formula - example Suppose, in the same example as the one before, the range of error (E) is acceptable at $4.00, sample size is reduced.

61. Copyright © 2000 by Harcourt, Inc. All rights reserved. Sample Size Formula - example

62.  1389  265.37 2  2 53.74 2        2 )29)(57.2( n 2         347   6325.18 2  4 53.74 2        4 )29)(57.2( n 2        99% Confidence Calculating Sample Size Copyright © 2000 by Harcourt, Inc. All rights reserved.

63. STANDARD ERROR OF THE PROPORTION

64. Copyright © 2000 by Harcourt, Inc. All rights reserved. CONFIDENCE INTERVAL FOR A PROPORTION

65. Copyright © 2000 by Harcourt, Inc. All rights reserved. SAMPLE SIZE FOR A PROPORTION

66. Copyright © 2000 by Harcourt, Inc. All rights reserved. 2 2 E pqz n  Where n = Number of items in samples Z 2 = The square of the confidence interval in standard error units. p = Estimated proportion of success q = (1-p) or estimated the proportion of failures E 2 = The square of the maximum allowance for error between the true proportion and sample proportion or zs p squared. The Sample Size Formula for a Proportion

67. Copyright © 2000 by Harcourt, Inc. All rights reserved. Calculating Sample Size at the 95% Confidence Level 753  001225. 922.  001225 )24)(.8416.3(  )035(. )4 )(. 6(.) 96 1. ( n 4.q 6.p 2 2   