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 (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 Frequency Distribution of Deposits

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

11. Population Mean

12. Sample Mean

13. Number of Salesperson Sales calls Mike 4 Patty 3 Billie 2 Bob 5 John 3 Frank 3 Chuck 1 Samantha 5 26 Number of Sales Calls Per Day by Salespersons

14. Product AProduct B 196150 198160 199176 199181 200192200 200201 201202 201213 201224 202240 202261 Sales for Products A and B, Both Average 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:

19. 150 160 170 180 190 200210 5432154321 Low Dispersion Value on Variable Frequency Low Dispersion Verses High Dispersion

20. 150 160 170 180 190 200210 5432154321 Frequency High dispersion Value on Variable Low Dispersion Verses High Dispersion

21. Average Deviation

22. Mean Squared Deviation

23. The Variance

24. Variance

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

26. Sample Standard Deviation

27. Population Standard Deviation

28. Sample Standard Deviation

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. MEAN Normal Distribution

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

33. 85115 100 14570 Normal Curve: IQ Example

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. 0 1 -2 2 z A Standardized Normal Curve

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

38. Standardized Scores

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

40. 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. Population distribution Sample distribution Sampling distribution

42.  x  Population Distribution

43.  X S Sample Distribution

44. Sampling Distribution

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

46. Central Limit Theorem

47. Standard Error of the Mean

49. Parameter Estimates Point estimates Confidence interval estimates

50. Confidence Interval

54. Estimating the Standard Error of the Mean

56. Random Sampling Error and Sample Size are Related

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

58. Sample Size Formula

59. 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.

60. Sample Size Formula - Example

61. Suppose, in the same example as the one before, the range of error (E) is acceptable at $4.00, sample size is reduced. Sample Size Formula - Example

63. 99% Confidence Calculating Sample Size  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       

64. Standard Error of the Proportion

65. Confidence Interval for a Proportion

66. Sample Size for a Proportion

67. 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.

68. 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   