Research Methods William G. Zikmund, Ch17

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

2. What does Statistics Mean? Descriptive statistics Number of people Trends in employment Data Inferential statistics Make an inference about a population from a sample

3. Population Parameter Versus Sample Statistics

4. Population Parameter Variables in a population Measured characteristics of a population Greek lower-case letters as notation

5. Sample Statistics Variables in a sample Measures computed from data English letters for notation

6. Making Data Usable Frequency distributions Proportions Central tendency Mean Median Mode Measures of dispersion

7. 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. Percentage Distribution of Amounts of DepositsAmountPercentless 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

9. Probability Distribution of Amounts of DepositsAmount Probabilityless than $3,000 .16$3,000 - $4,999 .17$5,000 - $9,999 .18$10,000 - $14,999 .23$15,000 or more .26

10. Measures of Central Tendency Mean - arithmetic average , Population; X , sample Median - midpoint of the distribution Mode - the value that occurs most often

11. Population Mean X = i N

12. Sample Mean XiX= n

13. 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. Sales for Products A and B, Both Average 200Product A Product B 196 150 198 160 199 176 199 181 200 192 200 200 200

15. Measures of Dispersion The range Standard deviation

16. Measures of Dispersion or Spread Range Mean absolute deviation Variance Standard deviation

17. 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. Deviation Scores The differences between each observation value and the mean: d x x i = i

19. Low Dispersion Verses High Dispersion 5 Low DispersionFrequency 4 3 2 1 150 160 170 180 190 200 210 Value on Variable

20. Low Dispersion Verses High Dispersion 5Frequency 4 High dispersion 3 2 1 150 160 170 180 190 200 210 Value on Variable

21. Average Deviation (X i X ) =0 n

22. Mean Squared Deviation ( Xi X ) 2 n

23. The VariancePopulation 2Sample 2S

24. Variance ( X X ) 2S = 2 n 1

25. Variance The variance is given in squared units The standard deviation is the square root of variance:

26. Sample Standard Deviation ( Xi X ) S= n1 2

27. Population Standard Deviation = 2

28. Sample Standard Deviation S= S 2

29. Sample Standard Deviation ( Xi X ) S= n1 2

30. The Normal Distribution Normal curve Bell shaped Almost all of its values are within plus or minus 3 standard deviations I.Q. is an example

31. Normal Distribution MEAN

32. Normal Distribution 13.59% 34.13% 34.13% 13.59% 2.14%2.14%

33. Normal Curve: IQ Example 70 85 100 115 145

34. Standardized Normal Distribution Symetrical about its mean Mean identifies highest point Infinite number of cases - a continuous distribution Area under curve has a probability density = 1.0 Mean of zero, standard deviation of 1

35. 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. A Standardized Normal Curve -2 -1 0 1 2 z

37. The Standardized Normal is the Distribution of Z z +z

38. Standardized Scores x z=

39. Standardized Values Used to compare an individual value to the population mean in units of the standard x deviation z=

40. Linear Transformation of Any Normal Variable Into a Standardized Normal Variable X Sometimes the Sometimes thescale is stretched scale is shrunk x z= -2 -1 0 1 2

41. Population distributionSample distributionSampling distribution

42. Population Distribution x

43. Sample Distribution _ S X

44. Sampling Distribution X SX X

45. Standard Error of the Mean Standard deviation of the sampling distribution

46. Central Limit Theorem

47. Standard Error of the Mean Sx = n

48. Distribution Mean Standard DeviationPopulation Sample S XSampling X SX

49. Parameter Estimates Point estimates Confidence interval estimates

50. Confidence Interval = X a small sampling error

51. SMALL SAMPLING ERROR = Z cl S X

52. E = Z cl S X

53. =X E

54. Estimating the Standard Error of the Mean S Sx = n

55. S = X Z cl n

56. Random Sampling Error and Sample Size are Related

57. Sample Size Variance (standard deviation) Magnitude of error Confidence level

58. Sample Size Formula 2 zs n= E

59. Sample Size Formula - ExampleSuppose a survey researcher, studyingexpenditures on lipstick, wishes to have a95 percent confident level (Z) and arange of error (E) of less than $2.00. Theestimate of the standard deviation is$29.00.

60. Sample Size Formula - Example (1.96 )( 29.00) 2 2 zs n= = E 2.00 2 56.84 = = ( 28.42 ) = 808 2 2.00

61. Sample Size Formula - ExampleSuppose, in the same example as the onebefore, the range of error (E) isacceptable at $4.00, sample size isreduced.

62. Sample Size Formula - Example ( 1.96)( 29.00) 2 2 zs n= = E 4.00 2 56.84 = = ( 14.21) = 202 2 4.00

63. Calculating Sample Size 99% Confidence 2 2 (2.57)(29) (2.57)(29) n= n= 2 4 2 2 74.53 74.53 = = 2 4 = [ .265] 37 2 = [ .6325 18 ] 2=1389 = 347

64. Standard Error of the Proportion sp = pq n or p ( 1p ) n

65. Confidence Interval for a Proportion pZ S cl p

66. Sample Size for a Proportion Z pq2 n= E 2

67. z2pq n= 2 EWhere: n = Number of items in samplesZ2 = The square of the confidence interval in standard error units. p = Estimated proportion of success q = (1-p) or estimated the proportion of failuresE2 = The square of the maximum allowance for error between the true proportion and sample proportion or zsp squared.

68. Calculating Sample Sizeat the 95% Confidence Level p = .6 ( 96 )2(. 6)(. 4 ) 1. n= q = .4 ( . 035 )2 (3. 8416)(. 24) = 001225 . 922 = . 001225 = 753

Research Methods William G. Zikmund, Ch17

Business

sample size formula

sample standard deviation

sample size

standard deviation

standard error

confidence interval

curve

distribution