Learning Objectives Copyright © 2002 South-Western/Thomson Learning Statistical Testing of...

Learning ObjectivesLearning Objectives

Copyright © 2002 South-Western/Thomson Learning

Statistical Testing of Differences

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Learning Objectives

1. To become aware of the nature of statistical significance.

2. To understand the concept of hypothesis development and how to test hypotheses.

3. To understand the differences between Type I and Type II errors.

4. To be familiar with several of the more common statistical tests of goodness of fit, hypotheses about one mean, hypothesis about two means, and hypotheses about proportions.

5. To learn about analysis of variance.


Are certain measures different from one another?

For example:

Did top-of-mind awareness really increase?

Did customer satisfaction really increase?

To become aware of the nature of statistical differences.

Differences and Changes


Statistical Significance

It is possible for numbers to be different in a mathematical sense but not statistically different in a statistical sense.

• Mathematical differences

• Statistical significance

• Managerially important differences

To become aware of the nature of statistical differences.


HypothesisAn assumption that a researcher makes about some characteristic of the population under study.

Steps in Hypothesis Testing

Step One: Stating the Hypothesis

Null hypothesis: Ho

Alternative hypothesis: Ha

Step Two: Choosing the Appropriate Test Statistic

To understand the concept of hypothesis development and how to test hypotheses.

Hypothesis Testing


Step Three: Developing a Decision Rule

Step Four: Calculating the Value of the Test Statistic

• Use the appropriate formula

• Compare calculated value to the critical value.

• State the result in terms of:

• rejecting the null hypothesis

• failing to reject the null hypothesis

Step Five: Stating the Conclusion

To understand the concept of hypothesis development and how to test hypotheses.

Hypothesis Testing


Types of Errors in Hypothesis Testing

Type I Error

Rejection of the null hypothesis when, in fact, it is true.

Type II Error

Acceptance of the null hypothesis when, in fact, it is false.

Accepting Ho or Failing to Reject Ho?

One-Tailed Test or Two-Tailed Test?

To understand the differences between Type I and Type II errors.Other issues

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Type I and Type II Errors

Actual State of the Null Hypothesis

Fail to Reject Ho Reject Ho

Ho is true

Ho is false

Correct (1-) no error

Type II error ()

Type I error ()

Correct (1- ) no error

Tab

le 1

5.2


Independent Versus Related Samples

Independent samples

Measurement of a variable in one population has no effect on the measurement of the other variable

Related Samples

Measurement of a variable in one population may influence the measurement of the other variable.

Degrees of Freedom

The number of observations minus the number of constraints.

To understand the concept of hypothesis development and testing a hypothesis.

Commonly UsedStatistical Hypothesis Tests


Chi-Square

To determine whether an observed pattern of frequencies corresponds to an “expected” pattern.

Chi-Square Test of a Single Sample

1. Specify the null and alternative hypotheses.

2. Determine the number of visitors that would be expected in each category if the null hypotheses were correct (Ei).

To describe statistical tests of goodness of fit.

Goodness of Fit


3. Calculate the X2 value using the formula:


Oi = observed number in i th category

Ei = expected number in i th category

k = number of categories

X2 k

I = 1 Ei

(Oi - Ei ) 2 =

Goodness of Fit


4. Select the level of significance .

5. Compare the calculated X2 value with the table value. State the result.


Chi-Square Test of Two Independent Samples

Association between two or more variables.

1. State the null and alternative hypotheses.

Goodness of Fit



2. Place the observed (sample) frequencies in a k x r table using the k columns for the sample groups and the r rows for the conditions or treatments.

Calculate the sum of each row and each column.

Record marginal totals.

Calculate total for entire table.

3. Determine the expected frequency for each cell in the contingency table by calculating the product of the two marginal totals common to that cell and dividing that value by N.

Goodness of Fit



4. Calculate the value of X2 using:

X2 k

j = 1 Eij

(Oij - E ij) 2 =

r

I = 1

Oij = observed number in i th row of the jth column

Eij = expected number in i th row of the jth column

5. Compare the calculated X2 with the tabular X2. State the result.

Goodness of Fit


Kolmogorov-Smirnov Test


2. Establish the cumulative frequency distribution expected under the null hypothesis.

3. Calculate the observed cumulative frequency distribution from the sample.

4. Select the level of significance .

5. Determine the K-S statistic.

6. State the result.


Goodness of Fit


Z - Test

Hypothesis tests about a single mean if the sample mean if the sample is large enough and drawn from a normal population.


2. Specify the level of sampling error () allowed.

3. Determine the sample standard deviation.

4. Calculate the estimated standard error of the mean:

To describe statistical tests about one mean.

About One Mean


Sx =S

n√where

Sx = the estimated standard error of the mean.


About One Mean


About One Mean

5. Calculate the test statistic:

(sample mean) population mean under the null hypothesis

estimated standard error of the mean

Z =




t - Test

Hypothesis tests about a single mean if the sample mean if the sample is too small to use the Z - test


2. Specify the level of sampling error () allowed.

3. Determine the sample standard deviation.


About One Mean


S

n

I = 1

n - 1

(Xi - X) 2 = √

where

Xi = observed sales per week in the ith store

X = average sales per week

n = the number of stores


About One Mean


4. Calculate the estimated standard error of the mean using:


About One Mean

Sx =S

n√


5. Calculate the t - statistic:

(sample mean) population mean under the null hypothesis

estimated standard error of the mean

t =


About One Mean



Marketers are frequently interested in testing differences between groups.

1. The null and alternative hypotheses

2. Set the level of sampling error ()

3. Estimate the standard error of the differences between two means:

About Two Means To describe statistical tests about two means.


Sx m-ƒ= S2

m S2 ƒ

nm n ƒ

+√Sm = estimated standard deviation of population m (men)

Sf = estimated standard deviation of population f (women)

nm = sample size for sample m

nf = sample size for sample f

To describe statistical tests about two means.

About Two Means


4. Calculate the test statistic.

Z is calculated as:

difference between means of first sample and second sample

difference between means under the null hypothesis

standard error or the differences between the meansZ =

5. Compare the calculated value of Z with the critical value. State the result.

To describe statistical tests about two means.

About Two Means


Test of a Proportion, One Sample

Phenomena expressed as percentages

1. Specify the null and alternative hypotheses


3. Estimate the standard error using the P-value specified in the null hypothesis:

About Proportions To describe statistical tests about proportions.


Sp P(1- P)

n - 1

=√where:

P = proportion specified in the null hypothesis

n = sample size

To describe statistical tests about proportions.

About Proportions


(observed proportion - proportion under null hypothesis)

estimated standard error (Sp)

Compare the calculated Z-value with the critical Z-value.

Z =


About Proportions

4. Calculate the test statistic as follows:


Tests of Differences Between Two Proportions, Independent Samples

Management may be interested in the difference between the proportions of people in two different groups that engage in a certain activity or have a certain characteristic.



3. Estimate the standard error of the differences between the two proportions:


About Proportions



About Proportions

where

P =nmPm + nfPf

nm + nf

Pm = proportion in sample m (men)

Pf = proportion in sample f (women)

nm = sample of size m

nf = sample of size f

Spm-ƒ = 1

nm n ƒ+P(1- P)

1√


4. Calculate the test statistic:

difference between observed proportions

difference between proportions under the null hypothesis

estimated standard error or the differences between the two means

Z =


About Proportions

5. Compare the calculated Z-value with the critical Z-value. State the result.


Analysis of Variance

Analysis of Variance (ANOVA)

Test for the differences among the means of two or more variables.


2. Calculate sum of squares among groups (SSA)

SSA j= 1

nj (Xi - Xt ) 2 =

To learn about analysis of variance.


3. Calculate the variation among the group means as measured by the mean sum of squares among groups (MSA):

sum of squares among groups (SSA)

degrees of freedom (d.f.)MSA =



where Degrees of freedom = number of groups (C) -1


4. Calculate the sum of squares within groups or sum of squared error (SSE):

SSE I = 1

= (Xij- Xj) 2 j = 1

c n j

5. Calculate the mean square error (MSE):

sum of squares within groups (SSE)

degrees of freedom (d.f.)MSE =




6. Calculate the F - statistic as follows:

F = MSA

MSE



p - VALUES AND SIGNIFICANCE TESTING

P- Value

The exact probability of getting a computed test statistic that was largely due to chance.

The smaller the p-value, the smaller the probability that the observed result occurred by chance.



• Differences and Changes

• Statistical Significance

• Hypothesis Testing

• Other issues

• Hypothesis Tests

• About One Mean

• About Two Means

• About Proportions

• Analysis of Variance

• Goodness of Fit

SUMMARY

Learning Objectives Copyright © 2002 South-Western/Thomson Learning Statistical Testing of...

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