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### Practical Statistics

Mean Comparisons

There are six statistics that will

answer90% of all questions!

- Descriptive
- Chi-square
- Z-tests
- Comparison of Means
- Correlation
- Regression

- A sample mean against a hypothesis.

- A sample mean against a hypothesis.
- Two sample means compared to each other.

- A sample mean against a hypothesis.
- Two sample means compared to each other.
- Two means within the same sample.

The standard error for means is:

Hence for one mean compared to a hypothesis:

Each t value comes with a certain degree

of freedom df = n - 1

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?

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.

up on the internet….

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

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

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.

- A sample mean against a hypothesis.
- Two sample means compared to each other.
- Two means within the same sample.

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.

Can you see why?

It is caused by a common mistake of

confusing the sampling distribution

with a the sample distribution.

(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.

- A sample mean against a hypothesis.
- Two sample means compared to each other.
- Two means within the same sample.

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.

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.

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.

Compares the means of two or more groups

by comparing the variance between groups

with the variance that exists within groups.

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

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

The probability distribution is dependent upon

the degrees of freedom between and the

degrees of freedom within.

Typical output looks like this:

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.

Is the measure of personality different between

men and women?

Two sample means compared to each other.

Is the measure of personality different between

men and women?

Two sample means compared to each other.

Do respondents like themselves better than the

service provider?

Two means within the same sample.

Do respondents like themselves better than the

service provider?

Two means within the same sample.

Is personality difference by perception of

service encounter?

More than two sample means compared to each other.

Is personality difference by perception of

service encounter?

More than two sample means compared to each other.

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