Practical Statistics

Mean Comparisons

There are six statistics that willanswer 90% of all questions!1. Descriptive2. Chi-square3. Z-tests4. Comparison of Means5. Correlation6. Regression

t-test and ANOVA are for the means of

interval and ratio scales

These are very common statistics….

William S. Gosset1876-1937 Published under the

name: Student

t-test come in three types:

1. A sample mean against a hypothesis.

t-test come in three types:

1. A sample mean against a hypothesis.2.Two sample means compared to each other.

t-test come in three types:

1. A sample mean against a hypothesis.2.Two sample means compared to each other.3.Two means within the same sample.

t-test

The standard error for means is:

SEn x

0

t-test

Hence for one mean compared to a hypothesis:

tx H

n

0

0

Each t value comes with a certain degreeof freedom df = n - 1

t-test

IQ has a mean of 100 and a standard deviation of15. Suppose a group of immigrants came intoLondon. A random sample of 400 of these

Immigrants found an average IQ of 98.

Does this group have an IQ below thepopulation average?

t-test

The test statistic looks like this:

t

98 10015

400

2

0 752 67

..

There are n – 1 = 399 degrees of freedom.

The results are printed out by a computer or lookedup on a t-test table.

The critical value for399 degrees offreedom is about 1.97.

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

Of course, we could look thisup on the internet….

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

t-test

Since the test was “one-tailed,” the critical valueof 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:

1. A sample mean against a hypothesis.2.Two sample means compared to each other.3.Two means within the same sample.

t-test

The standard error of the difference between two means looks like this:

SEn n

12

1

22

21 2

t-test

Therefore the test statistic would look like this:

tx x

n n

( ) ( )1 2 1 2

12

1

22

2

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

t-test

Usually this is simplified by looking at the differencebetween two samples; so that:

tx x

sn np

( )

( )

1 2

2

1 2

1 1

Where:

Sn S n S

n np2 1 1

22 2

2

1 2

1 1

2

( ) ( )

Suppose that a new product was test marketed inthe 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 kilogramsa year (sd = 20), while a sample of 400 in Japan used an average of 180 kilograms a year (sd = 25).

Test the hypothesize…..

H 0 1 2 0

The test would start be computing:

S p2

2 21 20 400 1 25

500 400 2

(500 ) ( )

= 500

t

( )

.200 180

5000 89

The results would be written as:(t(898) = 0.89, ns),

and the conclusion isthat there is no difference in the

consumption rate between the U.S. and Japanese customers.

t

( )

.200 180

5000 89

But this is wrong!

Can you see why?

It is caused by a common mistake ofconfusing the sampling distributionwith 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.

)4001

5001

)(500(

)180200(

t

t-test come in three types:

1. A sample mean against a hypothesis.2.Two sample means compared to each other.3.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/orwhen the same person or object is measured twice in the same sample.

Student T1 T2 d

Tom 89 90 1Jan 88 91 3Jason 87 86 -1Halley 90 90 0Bill 75 79 4

The measurement of interest is d.

H0 : Average of d = 0

That is… the average differencebetween test 1 and test 2 is zero.

t-test The sampling error for this t-test is:

SES

nSDD

2

Were d = score(2) – score(1)

t-test The t-test is:

tD

S D

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 groupsthat need to be compared.

The t-test cannot be utilized for two reason.1.The number of pairs becomes large.

Suppose there are more than two groupsthat need to be compared.

The t-test cannot be utilized for two reason.1.The number of pairs becomes large.2.The probability of t is no longer accurate.

Hence a new statistic is needed:

The F-test

Or

Analysis of Variance (ANOVA) R.A. Fisher1880-1962

The F-test

Compares the means of two or more groupsby comparing the variance between groupswith the variance that exists within groups.

F is the ratio of variance:

22

21

S

S

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

The F-test

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

The F-testThe probability distribution is dependent uponthe degrees of freedom between and thedegrees of freedom within.

The F-testTypical output looks like this:

In SPSS ANOVA looks like this:

Descriptives

overall

64 8.5781 1.83272 .22909 8.1203 9.0359 1.00 10.00

24 8.2917 1.60106 .32682 7.6156 8.9677 3.00 10.00

38 9.1053 .92384 .14987 8.8016 9.4089 7.00 10.00

6 8.0000 1.09545 .44721 6.8504 9.1496 7.00 10.00

132 8.6515 1.56773 .13645 8.3816 8.9215 1.00 10.00

1.00

2.00

3.00

4.00

Total

N Mean Std. Deviation Std. Error Lower Bound Upper Bound

95% Confidence Interval forMean

Minimum Maximum

ANOVA

overall

13.823 3 4.608 1.914 .131

308.147 128 2.407

321.970 131

Between Groups

Within Groups

Total

Sum ofSquares df Mean Square F Sig.

22

21

S

S

Service Encounter

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 Encounter

Is the measure of personality different between men and women?

Two sample means compared to each other.

Service Encounter

Is the measure of personality different between men and women?

Two sample means compared to each other.

Service Encounter

Is the measure of personality different between men and women?

Service Encounter

Do respondents like themselves better than the service provider?

Two means within the same sample.

Service Encounter

Do respondents like themselves better than the service provider?

Two means within the same sample.

Service Encounter

Is the measure of personality different between shopping times?

Service Encounter

Is personality difference by perception ofservice encounter?

More than two sample means compared to each other.

Service Encounter

Is personality difference by perception ofservice encounter?

More than two sample means compared to each other.