Descriptive Statistics Descriptive Statistics describe a set of data.
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Transcript of Practical Statistics Mean Comparisons. There are six statistics that will answer 90% of all...
![Page 1: Practical Statistics Mean Comparisons. There are six statistics that will answer 90% of all questions! 1. Descriptive 2. Chi-square 3. Z-tests 4. Comparison.](https://reader030.fdocuments.net/reader030/viewer/2022033103/56649dac5503460f94a9af49/html5/thumbnails/1.jpg)
Practical Statistics
Mean Comparisons
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There are six statistics that willanswer 90% of all questions!1. Descriptive2. Chi-square3. Z-tests4. Comparison of Means5. Correlation6. Regression
![Page 3: Practical Statistics Mean Comparisons. There are six statistics that will answer 90% of all questions! 1. Descriptive 2. Chi-square 3. Z-tests 4. Comparison.](https://reader030.fdocuments.net/reader030/viewer/2022033103/56649dac5503460f94a9af49/html5/thumbnails/3.jpg)
t-test and ANOVA are for the means of
interval and ratio scales
These are very common statistics….
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William S. Gosset1876-1937 Published under the
name: Student
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t-test come in three types:
1. A sample mean against a hypothesis.
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t-test come in three types:
1. A sample mean against a hypothesis.2.Two sample means compared to each other.
![Page 7: Practical Statistics Mean Comparisons. There are six statistics that will answer 90% of all questions! 1. Descriptive 2. Chi-square 3. Z-tests 4. Comparison.](https://reader030.fdocuments.net/reader030/viewer/2022033103/56649dac5503460f94a9af49/html5/thumbnails/7.jpg)
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.
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t-test
The standard error for means is:
SEn x
0
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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
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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?
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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.
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The critical value for399 degrees offreedom is about 1.97.
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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
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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.
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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.
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t-test
The standard error of the difference between two means looks like this:
SEn n
12
1
22
21 2
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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
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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
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Where:
Sn S n S
n np2 1 1
22 2
2
1 2
1 1
2
( ) ( )
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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
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The test would start be computing:
S p2
2 21 20 400 1 25
500 400 2
(500 ) ( )
= 500
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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.
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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.
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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
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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.
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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.
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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.
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H0 : Average of d = 0
That is… the average differencebetween test 1 and test 2 is zero.
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t-test The sampling error for this t-test is:
SES
nSDD
2
Were d = score(2) – score(1)
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t-test The t-test is:
tD
S D
The degrees of freedom = n - 1
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Examples can be found at these sites:
http://en.wikipedia.org/wiki/T-test
http://canhelpyou.com/statistics/tTestDependentSamples.html
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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.
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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.
![Page 34: Practical Statistics Mean Comparisons. There are six statistics that will answer 90% of all questions! 1. Descriptive 2. Chi-square 3. Z-tests 4. Comparison.](https://reader030.fdocuments.net/reader030/viewer/2022033103/56649dac5503460f94a9af49/html5/thumbnails/34.jpg)
Hence a new statistic is needed:
The F-test
Or
Analysis of Variance (ANOVA) R.A. Fisher1880-1962
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The F-test
Compares the means of two or more groupsby comparing the variance between groupswith the variance that exists within groups.
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F is the ratio of variance:
22
21
S
S
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http://controls.engin.umich.edu/wiki/index.php/Factor_analysis_and_ANOVA
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The F-test
http://www.statsoft.com/textbook/distribution-tables/
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The F-testThe probability distribution is dependent uponthe degrees of freedom between and thedegrees of freedom within.
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The F-testTypical output looks like this:
![Page 43: Practical Statistics Mean Comparisons. There are six statistics that will answer 90% of all questions! 1. Descriptive 2. Chi-square 3. Z-tests 4. Comparison.](https://reader030.fdocuments.net/reader030/viewer/2022033103/56649dac5503460f94a9af49/html5/thumbnails/43.jpg)
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
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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.
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Service Encounter
Is the measure of personality different between men and women?
Two sample means compared to each other.
![Page 46: Practical Statistics Mean Comparisons. There are six statistics that will answer 90% of all questions! 1. Descriptive 2. Chi-square 3. Z-tests 4. Comparison.](https://reader030.fdocuments.net/reader030/viewer/2022033103/56649dac5503460f94a9af49/html5/thumbnails/46.jpg)
Service Encounter
Is the measure of personality different between men and women?
Two sample means compared to each other.
![Page 47: Practical Statistics Mean Comparisons. There are six statistics that will answer 90% of all questions! 1. Descriptive 2. Chi-square 3. Z-tests 4. Comparison.](https://reader030.fdocuments.net/reader030/viewer/2022033103/56649dac5503460f94a9af49/html5/thumbnails/47.jpg)
Service Encounter
Is the measure of personality different between men and women?
![Page 48: Practical Statistics Mean Comparisons. There are six statistics that will answer 90% of all questions! 1. Descriptive 2. Chi-square 3. Z-tests 4. Comparison.](https://reader030.fdocuments.net/reader030/viewer/2022033103/56649dac5503460f94a9af49/html5/thumbnails/48.jpg)
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Service Encounter
Do respondents like themselves better than the service provider?
Two means within the same sample.
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Service Encounter
Do respondents like themselves better than the service provider?
Two means within the same sample.
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Service Encounter
Is the measure of personality different between shopping times?
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Service Encounter
Is personality difference by perception ofservice encounter?
More than two sample means compared to each other.
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Service Encounter
Is personality difference by perception ofservice encounter?
More than two sample means compared to each other.