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

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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

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

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

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

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, $3,000 - $4, $5,000 - $9, $10,000 - $14, $15,000 or more 811 3,120

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

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

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

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

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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

Copyright © 2000 by Harcourt, Inc. All rights reserved. Sales for Products A and B, Both Average 200 Product AProduct B

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

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

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

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

Copyright © 2000 by Harcourt, Inc. All rights reserved. Low Dispersion Verses High Dispersion Low Dispersion Value on Variable Frequency

Copyright © 2000 by Harcourt, Inc. All rights reserved Frequency High dispersion Value on Variable

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

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

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

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

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

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

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

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

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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

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.

Copyright © 2000 by Harcourt, Inc. All rights reserved. A STANDARDIZED NORMAL CURVE z

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

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

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

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

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

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

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

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

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

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

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PARAMETER ESTIMATES POINT ESTIMATES CONFIDENCE INTERVAL ESTIMATES

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ESTIMATING THE STANDARD ERROR OF THE MEAN

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RANDOM SAMPLING ERROR AND SAMPLE SIZE ARE RELATED

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

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.

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

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.

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

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

STANDARD ERROR OF THE PROPORTION

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

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

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