BusinessResearch Methods

William G. Zikmund

Chapter 17:

Determination of Sample Size

What does Statistics Mean?

• Descriptive statistics– Number of people– Trends in employment– Data

• Inferential statistics– Make an inference about a population from a

sample

Population Parameter Versus Sample Statistics

Population Parameter

• Variables in a population

• Measured characteristics of a population

• Greek lower-case letters as notation

Sample Statistics

• Variables in a sample

• Measures computed from data

• English letters for notation

Making Data Usable

• Frequency distributions

• Proportions

• Central tendency– Mean– Median– Mode

• Measures of dispersion

Frequency (number ofpeople 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

Amount Percentless 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

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

Measures of Central Tendency

• Mean - arithmetic average– µ, Population; , sample

• Median - midpoint of the distribution

• Mode - the value that occurs most often

X

Population Mean

NXi

nX

Xi

Sample Mean

Number ofSalesperson Sales calls

Mike 4Patty 3Billie 2Bob 5John 3Frank 3Chuck 1Samantha 5

26

Number of Sales Calls Per Day by Salespersons

Product A Product B196150198160199176199181200192200200200201201202201213201224202240202261

Sales for Products A and B, Both Average 200

Measures of Dispersion

• The range

• Standard deviation

Measures of Dispersion or Spread

• Range

• Mean absolute deviation

• Variance

• Standard deviation

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

Deviation Scores

• The differences between each observation value and the mean:

xxd ii

150 160 170 180 190 200 210

5

4

3

2

1

Low Dispersion

Value on Variable

Fre

quen

cyLow Dispersion Verses High

Dispersion

150 160 170 180 190 200 210

5

4

3

2

1

Fre

quen

cy High dispersion

Value on Variable

Low Dispersion Verses High Dispersion

Average Deviation

0)(

n

XX i

Mean Squared Deviation

n

XXi 2)(

The Variance

2

2

S

Sample

Population

Variance

1)2

2

nXX

S

Variance

• The variance is given in squared units

• The standard deviation is the square root of variance:

Sample Standard Deviation

1

2

n

XX iS

Population Standard Deviation

2

Sample Standard Deviation

2SS

Sample Standard Deviation

1

2

n

XX iS

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

MEAN

Normal Distribution

2.14%

13.59% 34.13% 34.13% 13.59%

2.14%

Normal Distribution

85 115100 14570

Normal Curve: IQ Example

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

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

01 -1-2 2 z

A Standardized Normal Curve

The Standardized Normal is the Distribution of Z

–z +z

xz

Standardized Scores

xz

Standardized Values

• Used to compare an individual value to the population mean in units of the standard deviation

Linear Transformation of Any Normal Variable Into a Standardized Normal Variable

-2 -1 0 1 2

Sometimes thescale is stretched

Sometimes thescale is shrunk

X

xz

•Population distribution

•Sample distribution

•Sampling distribution

x

Population Distribution

XS

Sample Distribution

XS XX

Sampling Distribution

Standard Error of the Mean

• Standard deviation of the sampling distribution

Central Limit Theorem

Standard Error of the Mean

nSx

Distribution Mean StandardDeviation

Population Sample

X S

SamplingX

XS

Parameter Estimates

• Point estimates

• Confidence interval estimates

error sampling small aX

Confidence Interval

Xcl SZ ERRORSAMPLING SMALL

XclSZ E

E X

Estimating the Standard Error of the Mean

n

SSx

n

SZX cl

Random Sampling Error and Sample Size are Related

Sample Size

• Variance (standard deviation)

• Magnitude of error• Confidence level

Sample Size Formula

2

Ezs

n

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.

2

E

zsn

2

00.2

00.2996.1

2

00.2

84.56

242.28 808

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.

Sample Size Formula - Example

2

E

zsn

2

00.4

00.2996.1

2

00.4

84.56

221.14 202

Sample Size Formula - Example

99% ConfidenceCalculating Sample Size

1389

265.372

253.74

2

2)29)(57.2(n

2

347 6325.18 2

453.74

2

4)29)(57.2(n

2

npp

or

npq

ps

)1(

Standard Error of the Proportion

pclSZp

Confidence Interval for a Proportion

2

2

EpqZ

n

Sample Size for a Proportion

2

2

Epqz

n

Where: n = Number of items in samples

Z2 = The square of the confidence interval in standard error units.

p = Estimated proportion of success

q = (1-p) or estimated the proportion of failures

E2 = The square of the maximum allowance for error between the true proportion and sample proportion or zsp squared.

Calculating Sample Size at the 95% Confidence Level

753001225.

922.

001225

)24)(.8416.3(

)035( .)4)(.6(.)96 1. (

n4.q

6.p2

2