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Practical Statistics. Mean Comparisons. There are six statistics that will answer 90% of all questions! Descriptive Chi-square Z-tests Comparison of Means Correlation Regression. t-test and ANOVA are for the means of interval and ratio scales

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Practical statistics

Practical Statistics

Mean Comparisons


There are six statistics that will

answer90% of all questions!

  • Descriptive

  • Chi-square

  • Z-tests

  • Comparison of Means

  • Correlation

  • Regression


t-test and ANOVA are for the meansof

intervaland ratio scales

These are very common statistics….


William S. Gosset

1876-1937

Published under the

name: Student


t-test come in three types:

  • A sample mean against a hypothesis.


t-test come in three types:

  • A sample mean against a hypothesis.

  • Two sample means compared to each other.


t-test come in three types:

  • A sample mean against a hypothesis.

  • Two sample means compared to each other.

  • Two means within the same sample.


t-test

The standard error for means is:


t-test

Hence for one mean compared to a hypothesis:

Each t value comes with a certain degree

of freedom df = n - 1


t-test

IQ has a mean of 100 and a standard deviation of

15. Suppose a group of immigrants came into

London. A random sample of 400 of these

Immigrants found an average IQ of 98.

Does this group have an IQ below the

population average?


t-test

The test statistic looks like this:

There are n – 1 = 399 degrees of freedom.

The results are printed out by a computer or looked

up on a t-test table.


The critical value for

399 degrees of

freedom is about 1.97.


Of course, we could look this

up on the internet….

http://www.danielsoper.com/statcalc/calc08.aspx

For the IQ test: t(399) = 2.67, p = 0.00395


t-test

Since the test was “one-tailed,” the critical value

of t would be -1.65.

Therefore, t(399) = -2.67 would indicate

that the immigrants IQ is below normal.


t-test come in three types:

  • A sample mean against a hypothesis.

  • Two sample means compared to each other.

  • Two means within the same sample.


t-test

The standard error of the difference

between two means looks like this:


t-test

Therefore the test statistic would look like this:

With degrees of freedom = n(1) + n(2) - 2


t-test

Usually this is simplified by looking at the difference

between two samples; so that:



Suppose that a new product was test marketed in

the United States and in Japan. The company

hypothesizes that customers in both countries

would consume the product at the same rate.

A sample of 500 in the U.S. used an average of 200 kilograms

a year (sd = 20), while a sample of 400 in Japan used an

average of 180 kilograms a year (sd = 25).

Test the hypothesize…..



The results would be written as:

(t(898) = 0.89, ns),

and the conclusion is

that there is no difference in the consumption rate between the U.S. and Japanese customers.


But this is wrong!

Can you see why?

It is caused by a common mistake of

confusing the sampling distribution

with a the sample distribution.


The results are written as:

(t(898) = 13.33, p < .0001),

and the conclusion is that there is a large difference in the consumption rate between the U.S. and Japanese customers.


t-test come in three types:

  • A sample mean against a hypothesis.

  • Two sample means compared to each other.

  • Two means within the same sample.


t-test come in three types:

3. Two means within the same sample.

This t-test is used with correlated samples and/or

when the same person or object is measured

twice in the same sample.


Student T1 T2 d

Tom 89 90 1

Jan 88 91 3

Jason 87 86 -1

Halley 90 90 0

Bill 75 79 4

The measurement of interest is d.


H0 : Average of d = 0

That is… the average difference

between test 1 and test 2 is zero.


t-test

The sampling error for this t-test is:

Were d = score(2) – score(1)


t-test

The t-test is:

The degrees of freedom = n - 1


Examples can be found at these sites:

http://en.wikipedia.org/wiki/T-test

http://canhelpyou.com/statistics/tTestDependentSamples.html


Suppose there are more than two groups

that need to be compared.

The t-test cannot be utilized for two reason.

  • The number of pairs becomes large.


Suppose there are more than two groups

that need to be compared.

The t-test cannot be utilized for two reason.

  • The number of pairs becomes large.

  • The probability of t is no longer accurate.


Hence a new statistic

is needed:

The F-test

Or

Analysis of Variance (ANOVA)

R.A. Fisher

1880-1962


The F-test

Compares the means of two or more groups

by comparing the variance between groups

with the variance that exists within groups.


F is the ratio of variance:


http://controls.engin.umich.edu/wiki/index.php/Factor_analysis_and_ANOVAhttp://controls.engin.umich.edu/wiki/index.php/Factor_analysis_and_ANOVA


The F-test http://controls.engin.umich.edu/wiki/index.php/Factor_analysis_and_ANOVA

http://www.statsoft.com/textbook/distribution-tables/


The F-testhttp://controls.engin.umich.edu/wiki/index.php/Factor_analysis_and_ANOVA

The probability distribution is dependent upon

the degrees of freedom between and the

degrees of freedom within.


The F-testhttp://controls.engin.umich.edu/wiki/index.php/Factor_analysis_and_ANOVA

Typical output looks like this:


In SPSS http://controls.engin.umich.edu/wiki/index.php/Factor_analysis_and_ANOVAANOVA looks like this:


Service Encounterhttp://controls.engin.umich.edu/wiki/index.php/Factor_analysis_and_ANOVA

The average age of Iowans over 18 is

approximately 47. Is the sample a cross-section

of this population by age?

A sample mean against a hypothesis.


Service Encounterhttp://controls.engin.umich.edu/wiki/index.php/Factor_analysis_and_ANOVA

Is the measure of personality different between

men and women?

Two sample means compared to each other.


Service Encounterhttp://controls.engin.umich.edu/wiki/index.php/Factor_analysis_and_ANOVA

Is the measure of personality different between

men and women?

Two sample means compared to each other.


Service Encounterhttp://controls.engin.umich.edu/wiki/index.php/Factor_analysis_and_ANOVA

Is the measure of personality different between

men and women?


Service Encounterhttp://controls.engin.umich.edu/wiki/index.php/Factor_analysis_and_ANOVA

Do respondents like themselves better than the

service provider?

Two means within the same sample.


Service Encounterhttp://controls.engin.umich.edu/wiki/index.php/Factor_analysis_and_ANOVA

Do respondents like themselves better than the

service provider?

Two means within the same sample.


Service Encounterhttp://controls.engin.umich.edu/wiki/index.php/Factor_analysis_and_ANOVA

Is the measure of personality different between

shopping times?


Service Encounterhttp://controls.engin.umich.edu/wiki/index.php/Factor_analysis_and_ANOVA

Is personality difference by perception of

service encounter?

More than two sample means compared to each other.


Service Encounterhttp://controls.engin.umich.edu/wiki/index.php/Factor_analysis_and_ANOVA

Is personality difference by perception of

service encounter?

More than two sample means compared to each other.


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