1. Practical Statistics
Mean Comparisons

2. There are six statistics that will answer 90% of all questions!
Descriptive
Chi-square
Z-tests
Comparison of Means
Correlation
Regression

3. interval and ratio scales
t-test and ANOVA are for the means of
interval and ratio scales
They are very common statistics….

4. T-test
Why is it called a
t-test?

5. William S. Gosset
Published under the
name: Student

6. t-test come in three types:
A sample mean against a hypothesis.

7. t-test come in three types:
A sample mean against a hypothesis.
Two sample means compared to each other.

8. 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.

9. t-test
The standard error for means is:

10. Each t value comes with a certain degree
t-test
Hence for one mean compared to a hypothesis:
Each t value comes with a certain degree
of freedom df = n - 1

11. IQ has a mean of 100 and a standard deviation of
t-test
IQ has a mean of 100 and a standard deviation of
15. Suppose a group of immigrants came into
Iowa. A sample of 400 of these immigrants
found an average IQ of 98.
Does this group have
an IQ below the population average?

12. The test statistic looks like this:
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.

13. The critical value for
399 degrees of
freedom is about 1.97.

14. Of course, we could look this
up on the internet….
For the IQ test: t(399) = 2.67, p =

15. 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.

16. 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.

17. t-test
The standard error of the difference
between two means looks like this:

18. t-test
Therefore the test statistic would look like this:
With degrees of freedom = n(1) + n(2) - 2

19. t-test
Usually this is simplified by looking at the
difference between two samples; so that:

20. Where:

22. 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…..

23. The test would start be computing:
= (a SD = 22.36)

24. 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.

25. 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. 25

26. 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. 26

27. 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.

28. 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.

29. Student T1 T2 d Tom 89 90 1 Jan 88 91 3 Jason 87 86 -1 Halley 90 90 0
Bill
The measurement of interest is d.

30. That is… the average difference between test 1 and test 2 is zero.
H0 : Average of d = 0
That is… the average difference
between test 1 and test 2 is zero.

31. t-test
The sampling error for this t-test is:
Were d = score(2) – score(1)

32. t-test
The t-test is:
The degrees of freedom = n - 1

33. Examples can be found at these sites:

34. 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.

35. Analysis of Variance (ANOVA)
Hence a new statistic
is needed:
The F-test Or
Analysis of Variance (ANOVA)
R.A. Fisher

36. Compares the means of two or more groups
The F-test
Compares the means of two or more groups
by comparing the variance between groups
with the variance that exists within groups.
According to the Central Limit Theorem there is a relationship
between the variance of a statistic and the variance of the
population. If that relationship is violated, it is likely that the
statistics did not come from the same population as the other
statistics.

37. https://en.wikipedia.org/wiki/Analysis_of_variance

38. F is the ratio of variance:

40. The F-test

41. The F-test
The probability distribution is dependent upon
the degrees of freedom between and the
degrees of freedom within.

42. The F-test
Typical output looks like this:

43. In SPSS ANOVA looks like this: