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# Research Methods William G. Zikmund, Ch17

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Research Methods
William G. Zikmund

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Statistics
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• ### Research Methods William G. Zikmund, Ch17

1. 1. Business Research Methods William G. Zikmund Chapter 17:Determination of Sample Size
2. 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. 3. Population Parameter Versus Sample Statistics
4. 4. Population Parameter• Variables in a population• Measured characteristics of a population• Greek lower-case letters as notation
5. 5. Sample Statistics• Variables in a sample• Measures computed from data• English letters for notation
6. 6. Making Data Usable• Frequency distributions• Proportions• Central tendency – Mean – Median – Mode• Measures of dispersion
7. 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. 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. 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. 10. Measures of Central Tendency• Mean - arithmetic average – µ, Population; X , sample• Median - midpoint of the distribution• Mode - the value that occurs most often
11. 11. Population Mean ΣX µ= i N
12. 12. Sample Mean Σ XiX= n
13. 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. 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. 15. Measures of Dispersion• The range• Standard deviation
16. 16. Measures of Dispersion or Spread• Range• Mean absolute deviation• Variance• Standard deviation
17. 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. 18. Deviation Scores• The differences between each observation value and the mean: d x x i = i −
19. 19. Low Dispersion Verses High Dispersion 5 Low DispersionFrequency 4 3 2 1 150 160 170 180 190 200 210 Value on Variable
20. 20. Low Dispersion Verses High Dispersion 5Frequency 4 High dispersion 3 2 1 150 160 170 180 190 200 210 Value on Variable
21. 21. Average Deviation∑ (X i − X ) =0 n
22. 22. Mean Squared Deviation∑ ( Xi − X ) 2 n
23. 23. The VariancePopulationσ 2Sample 2S
24. 24. Variance Σ( X − X ) 2S = 2 n −1
25. 25. Variance• The variance is given in squared units• The standard deviation is the square root of variance:
26. 26. Sample Standard Deviation Σ ( Xi − X ) S= n−1 2
27. 27. Population Standard Deviation σ= σ 2
28. 28. Sample Standard Deviation S= S 2
29. 29. Sample Standard Deviation Σ ( Xi − X ) S= n−1 2
30. 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. 31. Normal Distribution MEAN
32. 32. Normal Distribution 13.59% 34.13% 34.13% 13.59% 2.14%2.14%
33. 33. Normal Curve: IQ Example 70 85 100 115 145
34. 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. 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. 36. A Standardized Normal Curve -2 -1 0 1 2 z
37. 37. The Standardized Normal is the Distribution of Z –z +z
38. 38. Standardized Scores x−µ z= σ
39. 39. Standardized Values• Used to compare an individual value to the population mean in units of the standard x−µ deviation z= σ
40. 40. Linear Transformation of Any Normal Variable Into a Standardized Normal Variable σ σ µ µ X Sometimes the Sometimes thescale is stretched scale is shrunk x−µ z= -2 -1 0 1 2 σ
41. 41. •Population distribution•Sample distribution•Sampling distribution
42. 42. Population Distribution −σ µ σ x
43. 43. Sample Distribution _ Χ S X
44. 44. Sampling Distribution µX SX X
45. 45. Standard Error of the Mean• Standard deviation of the sampling distribution
46. 46. Central Limit Theorem
47. 47. Standard Error of the Mean σ Sx = n
48. 48. Distribution Mean Standard DeviationPopulation µ σSample S XSampling µX SX
49. 49. Parameter Estimates• Point estimates• Confidence interval estimates
50. 50. Confidence Intervalµ = X ± a small sampling error
51. 51. SMALL SAMPLING ERROR = Z cl S X
52. 52. E = Z cl S X
53. 53. µ=X ±E
54. 54. Estimating the Standard Error of the Mean S Sx = n
55. 55. Sµ = X ± Z cl n
56. 56. Random Sampling Error and Sample Size are Related
57. 57. Sample Size• Variance (standard deviation)• Magnitude of error• Confidence level
58. 58. Sample Size Formula 2  zs  n=  E
59. 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. 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. 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. 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. 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. 64. Standard Error of the Proportion sp = pq n or p ( 1−p ) n
65. 65. Confidence Interval for a Proportion p±Z S cl p
66. 66. Sample Size for a Proportion Z pq2 n= E 2
67. 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. 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
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