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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


Practical statistics

There are six statistics that will

answer90% of all questions!

  • Descriptive

  • Chi-square

  • Z-tests

  • Comparison of Means

  • Correlation

  • Regression


Practical statistics

t-test and ANOVA are for the meansof

intervaland ratio scales

These are very common statistics….


Practical statistics

William S. Gosset

1876-1937

Published under the

name: Student


Practical statistics

t-test come in three types:

  • A sample mean against a hypothesis.


Practical statistics

t-test come in three types:

  • A sample mean against a hypothesis.

  • Two sample means compared to each other.


Practical statistics

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.


Practical statistics

t-test

The standard error for means is:


Practical statistics

t-test

Hence for one mean compared to a hypothesis:

Each t value comes with a certain degree

of freedom df = n - 1


Practical statistics

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?


Practical statistics

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.


Practical statistics

The critical value for

399 degrees of

freedom is about 1.97.


Practical statistics

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


Practical statistics

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.


Practical statistics

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.


Practical statistics

t-test

The standard error of the difference

between two means looks like this:


Practical statistics

t-test

Therefore the test statistic would look like this:

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


Practical statistics

t-test

Usually this is simplified by looking at the difference

between two samples; so that:



Practical statistics

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…..



Practical statistics

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.


Practical statistics

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.


Practical statistics

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.


Practical statistics

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.


Practical statistics

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.


Practical statistics

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.


Practical statistics

H0 : Average of d = 0

That is… the average difference

between test 1 and test 2 is zero.


Practical statistics

t-test

The sampling error for this t-test is:

Were d = score(2) – score(1)


Practical statistics

t-test

The t-test is:

The degrees of freedom = n - 1


Practical statistics

Examples can be found at these sites:

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

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


Practical statistics

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.


Practical statistics

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.


Practical statistics

Hence a new statistic

is needed:

The F-test

Or

Analysis of Variance (ANOVA)

R.A. Fisher

1880-1962


Practical statistics

The F-test

Compares the means of two or more groups

by comparing the variance between groups

with the variance that exists within groups.


Practical statistics

F is the ratio of variance:


Practical statistics

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


Practical statistics

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

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


Practical statistics

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.


Practical statistics

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

Typical output looks like this:


Practical statistics

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


Practical statistics

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.


Practical statistics

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.


Practical statistics

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.


Practical statistics

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

Is the measure of personality different between

men and women?


Practical statistics

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.


Practical statistics

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.


Practical statistics

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

Is the measure of personality different between

shopping times?


Practical statistics

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.


Practical statistics

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.