business research methods william g. zikmund chapter 21: univariate statistics

Business Research Methods

William G. Zikmund

Chapter 21:

Univariate Statistics

Copyright © 2000 by Harcourt, Inc.

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

UNIVARIATE STATISTICS

• TEST OF STATISTICAL SIGNIFICANCE

• HYPOTHESIS TESTING ONE VARIABLE AT A TIME


HYPOTHESIS

• UNPROVEN PROPOSITION

• SUPPOSITION THAT TENATIVELY EXPLAINS CERTAIN FACTS OR PHENOMONA

• ASSUMPTION ABOUT NATURE OF THE WORLD


HYPOTHESIS

• AN UNPROVEN PROPOSITION OR SUPPOSITION THAT TENTATIVELY EXPLAINS CERTAIN FACTS OF PHENOMENA

• NULL HYPOTHESIS

• ALTERNATIVE HYPOTHESIS


NULL HYPOTHESIS

• STATEMENT ABOUT THE STATUS QUO

• NO DIFFERENCE


ALTERNATIVE HYPHOTESIS

• STATEMENT THAT INDICATES THE OPPOSITE OF THE NULL HYPOTHESIS


SIGNIFICANCE LEVEL

• CRITICAL PROBABLITY IN CHOOSING BETWEEN THE NULL HYPOTHESIS AND THE ALTERNATIVE HYPOTHESIS


SIGNIFICANCE LEVEL

• CRITICAL PROBABLITY

• CONFIDENCE LEVEL

• ALPHA

• PROBABLITY LEVEL SELECTED IS TYPICALLY .05 OR .01

• TOO LOW TO WARRANT SUPPORT FOR THE NULL HYPOTHESIS


0.3 : oH

The null hypothesis that the mean is equal to 3.0:


0.3 :1 H

The alternative hypothesis that the mean does not equal to 3.0:


A SAMPLING DISTRIBUTION

x


A SAMPLING DISTRIBUTION

LOWER LIMIT

UPPERLIMIT


Critical values of

Critical value - upper limit

n

SZZS X or

225

5.196.1 0.3


Critical values of

1.096.1 0.3

196. 0.3

196.3


Critical values of

Critical value - lower limit

n

SZZS

X- or -

225

5.196.1- 0.3


Critical values of

1.096.1 0.3

196. 0.3

804.2


REGION OF REJECTION

LOWER LIMIT

UPPERLIMIT


HYPOTHESIS TEST

2.804 3.196 3.78


TYPE I AND TYPE II ERRORS

Accept null Reject null

Null is true

Null is false

Correct-Correct-no errorno error

Type IType Ierrorerror

Type IIType IIerrorerror

Correct-Correct-no errorno error


Type I and Type II Errors in Hypothesis Testing

State of Null Hypothesis Decisionin the Population Accept Ho Reject Ho

Ho is true Correct--no error Type I errorHo is false Type II error Correct--no error


CALCULATING ZOBS

xs

xzOBS


Alternate way of testing the hypothesis

X

obs S

XZ


Alternate way of testing the hypothesis

X

obs SZ

78.3

1.

0.378.3

1.

78.0 8.7


CHOOSING THE APPROPRAITE STATISTICAL TECHNIQUE

• Type of question to be answered

• Number of variables– Univariate– Bivariate– Multivariate

• Scale of measurement


PARAMETRICSTATISTICS

NONPARAMETRICSTATISTICS


t-distribution

• Symmetrical, bell-shaped distribution

• Mean of zero and a unit standard deviation

• Shape influenced by degrees of freedom


DEGREES OF FREEDOM

• Abbreviated d.f.

• Number of observations

• Number of constraints


or

Confidence interval estimate using the t-distribution

Xlc StX ..

n

StX lc ..limitUpper

n

StX lc ..limitLower


= population mean

= sample mean

= critical value of t at a specified confidence

level

= standard error of the mean

= sample standard deviation

= sample size

Confidence interval estimate using the t-distribution

..lct

X

XSSn


Confidence Interval using t

xcl stX

17

66.2

7.3

n

S

X


07.5

)1766.2(12.27.3limitupper


33.2

)1766.2(12.27.3limitLower


HYPOTHESIS TEST USING THE t-DISTRIBUTION


Univariate hypothesis test utilizing the t-distribution

Suppose that a production manager believes the average number of defective assemblies each day to be 20. The factory records the number of defective assemblies for each of the 25 days it was opened in a given month. The mean was calculated to be 22, and the standard deviation, ,to be 5.

XS


20 :

20 :

1

0

H

H


nSS X /25/5

1


Univariate hypothesis test utilizing the t-distribution

The researcher desired a 95 percent confidence, and the significance level becomes .05.The researcher must then find the upper and lower limits of the confidence interval to determine the region of rejection. Thus, the value of t is needed. For 24 degrees of freedom (n-1, 25-1), the t-value is 2.064.


:limitLower 25/5064.220 .. Xlc St 1064.220

936.17


:limitUpper 25/5064.220 ..

Xlc St 1064.220

064.20


Univariate hypothesis test - t-test

X

obs S

Xt

1

2022

1

2

2


TESTING A HYPOTHESIS ABOUT A DISTRIBUTION

• CHI-SQUARE TEST

• TEST FOR SIGNIFANCE IN THE ANALYSIS OF FREQUENCY DISTRIBUTIONS

• COMPARE OBSERVED FREQUENCIES WITH EXPECTED FREQUENCIES

• “GOODNESS OF FIT”


i

ii )²( ²

E

EOx

Chi-Square Test


Chi-Square Test

x² = chi-square statisticsOi = observed frequency in the ith cellEi = expected frequency on the ith cell


n

CRE ji

ij

Chi-Square Test - estimation for expected number for each cell


Chi-Square Test - estimation for expected number for each cell

Ri = total observed frequency in the ith rowCj = total observed frequency in the jth columnn = sample size


Univariate hypothesis test - Chi-square Example

2

222

1

2112

E

EO

E

EOX


Univariate hypothesis test - Chi-square Example

50

5040

50

5060 222

X

4


HYPOTHESIS TEST OF A PROPORTION

is the population proportion

p is the sample proportion

is estimated with p


Hypothesis Test of a Proportion

5. :H

5. :H

1

0


100

4.06.0pS

100

24.

0024. 04899.


pS

pZobs

04899.

5.6.

04899.

1. 04.2


0115.Sp

000133.Sp 1200

16.Sp

1200

)8)(.2(.Sp

n

pqSp

20.p 200,1n

Hypothesis Test of a Proportion: Another Example


Indeed .001 the beyond t significant is it

level. .05 the at rejected be should hypothesis null the so 1.96, exceeds value Z The

348.4Z0115.05.

Z

0115.15.20.

Z

Sp

Zp

Hypothesis Test of a Proportion: Another Example

business research methods william g. zikmund chapter 21: univariate statistics

Documents

null hypothesis slide

error slide

world slide

time slide

difference slide

critical values of slide

sampling distribution

null hypothesis statement