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Research Methods

William G. Zikmund

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- 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 – µ, Population; X , sample• Median - midpoint of the distribution• Mode - the value that occurs most often
- 11. Population Mean ΣX µ= i N
- 12. Sample Mean Σ 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 VariancePopulationσ 2Sample 2S
- 24. Variance Σ( 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 Σ ( Xi − X ) S= n−1 2
- 27. Population Standard Deviation σ= σ 2
- 28. Sample Standard Deviation S= S 2
- 29. Sample Standard Deviation Σ ( Xi − X ) S= n−1 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−µ z= σ
- 39. Standardized Values• Used to compare an individual value to the population mean in units of the standard x−µ deviation z= σ
- 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. •Population distribution•Sample distribution•Sampling distribution
- 42. Population Distribution −σ µ σ x
- 43. Sample Distribution _ Χ S X
- 44. Sampling Distribution µX SX X
- 45. Standard Error of the Mean• Standard deviation of the sampling distribution
- 46. Central Limit Theorem
- 47. Standard Error of the Mean σ Sx = n
- 48. Distribution Mean Standard DeviationPopulation µ σSample S XSampling µX SX
- 49. Parameter Estimates• Point estimates• Confidence interval estimates
- 50. Confidence Intervalµ = X ± a small sampling error
- 51. SMALL SAMPLING ERROR = Z cl S X
- 52. E = Z cl S X
- 53. µ=X ±E
- 54. Estimating the Standard Error of the Mean S Sx = n
- 55. Sµ = 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 ( 1−p ) n
- 65. Confidence Interval for a Proportion p±Z 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

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