1. Practical Statistics
Abbreviated Summary

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

3. Z-test are for proportions.
This test is so easy….
That it is not even given in some computer
programs like SPSS…..

4. Z-test are for proportions.
What is the probability that out of 250
customers, 220 would like the service
when the usual percent is that 70%
(175 out of 250) are satisfied?

5. Z-test are for proportions.
What is the probability that out of a
random sample of male and female
customers, the percent of both men and
women who like a new product is the
same?

6. Z-test are for proportions.
They come in two types:
A sample proportion against a hypothesis.

7. Z-test are for proportions.
They come in two types:
A sample proportion against a hypothesis.
Two samples compared to each other.

8. Z-test are for proportions.
The standard error (sampling error) for
proportions is:
Where p = freq/total and q = 1 - p

9. Z-test are for proportions.
Hence:
Where p is the hypothesized value, and pt is
the proportion found in a sample of size n.

10. Z-test are for proportions.
Suppose that XYZ Company believed that 20%
of their customers bought 80% of their product
(“heavy half”).
A sample of 200 customers found that 25% bought
80% of the product. Was the company correct in
their estimate?

11. Z-test are for proportions.
The test statistic looks like this:

12. Z-test are for proportions.
Since the test was “two-tailed,” the critical
value of Z would be Therefore, we would
conclude that there is not enough evidence to
over-ride the assumption that 20% of the
customers bought 80% of the product.

13. Z-test are for proportions.
P = 0.077

14. Z-test are for proportions.
They come in two types:
A sample proportion against a hypothesis.
Two samples compared to each other.

15. Z-test are for proportions.
The test for this case looks like this:

16. Z-test are for proportions.
Usually, the test assumes that the two groups
Are equal, or:

17. There is a problem here.
What is the value of p: ?

18. p is the value of the population
proportion, but we usually don’t know that,
so p is estimated by the weighted average
of the two groups….

19. Suppose that a new product was test marketed in the United States and
in Japan. The company hypothesizes that both countries response to the
product will be the same.
80% of a sample of 500 said they would buy the product again in the U.S.,
while 75% of a sample of 200 in Japan said they would buy the product
again.

20. Test the hypothesize…..

21. Since p = 0.80 in the U.S., and 0.75 in Japan,
the weighted average is used for p.
So:
p = ((.8 x 500)+(.75 x 200))/700 = 0.786

22. The test would be:
Z = .05/ = The critical value is 1.96; p =
The null hypothesis cannot be rejected, the U.S. and Japanese
customers are assumed to be the same.

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

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

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

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

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

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

29. 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.

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

31. 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

32. 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?

33. 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.

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

35. Therefore, t(399) = 2.67 would indicate
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.

36. 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.

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

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

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

40. Where:

42. 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…..

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

44. 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. 44

45. 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.

46. 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.

47. 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.

48. 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.

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

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

51. 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.

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

53. 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.

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

55. F is the ratio of variance:

57. The F-test

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

59. In SPSS ANOVA looks like this:

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

61. Correlation tests the degree of association
between interval and ratio measures.

62. Karl Pearson
Darwin
Galton

63. Correlation is based on a very simple idea that
Karl Pearson saw….

64. If you take two measures of the
same person or object, multiple them, and then add
the products across persons or objects… such as:
Person M M2 Product
Sum = 55

65. This is called: the sum of the cross products.
Person M M2 Product
Sum = 55

66. The largest possible sum will occur if M1 and M2
are in perfect ordinal order.
Note what happens when only one measure changes.
Person M M2 Product
Sum = 54

67. The smallest possible sum will occur if M1 and M2
are in perfect inverse order.
Person M M2 Product
Sum = 35

68. The “normal score” or
“standardized score” is equal to:

69. Converting the measures to Z scores…
Person M M2 Product
Sum = 5.0

70. Note, that the sum is equal to the number of people
in the sample, i.e., 5.
Person M M2 Product
Sum = 5.0

71. Note now what happens when the measure in Z scores
are arranging for perfect inverse order:
Person M M2 Product
Sum =

72. This is called “the sum of the cross products”

73. When X and Y are in ranked order the
So:
When X and Y are in ranked order the
max will be equal to n, and when ranked in
perfect negative order, the max will be -n.
(Or, n-1 and –(n-1) if taken from a sample).

74. The average of the sum of cross products is
the correlation.

75. r = 1.0 or r = -1.0 This means that a perfect association
always has a value of 1.0 when in positive order
and a value of -1.0 when in negative order.
A value of zero would indicate a random
relationship between the two variables.
r = 1.0 or r = -1.0

76. The correlation can be graphically shown by
using a scatter plot:

77. The correlation is related to the shape of
the scatter plot:

78. The correlation is an INDEX of association. It
contains three pieces of information:

79. The correlation is an INDEX of association. It
contains three pieces of information:
How much association is present, (an index)

80. The correlation is an INDEX of association. It
contains three pieces of information:
How much association is present,
Is that a “significant” association,
That is, can we reject the H0 that the
true association is zero.

81. The correlation is an INDEX of association. It
contains three pieces of information:
How much association is present,
Is that a “significant” association,
And, what is the magnitude of that association.

82. is the amount of variation accounted
The correlation is an INDEX of association. It
contains three pieces of information:
The correlation is r…..
and
r-squared
is the amount of variation accounted
for in Y by knowing X.

83. Be careful!! The correlation does not tell you that X is
the cause of Y.
It is a necessary condition for cause, but it does
not prove cause…

84. cum hoc ergo propter hoc
That correlation proves causation, is a logical fallacy by which two events that occur together are claimed to have a cause-and-effect relationship.
The fallacy is also known as
cum hoc ergo propter hoc
(Latin for "with this, therefore because of this")
and false cause.

86. The number of people waiting for a bus or train is highly
correlated with how long a person
must wait for a ride.

87. Trains come sooner
when more people
are waiting for it!
Does this mean that
It will come sooner
if you bring your
friends?

88. Hence, atmospheric CO2 causes crime.
Since the 1950s, both the atmospheric CO2 level and crime levels have increased sharply.
Hence, atmospheric CO2 causes crime.

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

91. Regression tests the degree of association
between interval and ratio measures,
AND
gives the best fit to the data.

92. Regression
Does three things:
1. Association
2. Best fit
3. Prediction

93. Regression
Regression creates an equation:
A simple linear equation would be:
Y = bX + a

94. Can we use the correlations to create equations
to estimate one variable from another?

95. For example:
Evaluations = b*Personality + a
Y = bX + a

96. So… Evaluation = 0.637 * Personality - 0.530

98. The equations do not have to be linear?

99. Regression can use more than one variable to
predict. This is called multiple regression.

100. When all these variables are put together, as they are in the real world,
only the instructors gender, what section a student took, the students’
GPA, and the evaluation the students gave to the instructor were related
to the final grade.

101. Path Diagram

102. Question:
What predicts the evaluation a class
and instructor will get?

104. The final evaluation of the class
and instructor is related to (in order):
Expected grade in Week 16
Actual grade in Week 16
Final grade for the class
Note: If all these grades were the same thing, only one would
be related.

105. Why? If we add variables one at a time,
sometimes the answer is different.
Notice below how the deserved grade at Week 16
becomes important.
Why?

106. This diagram shows that it is the expected grade
at Week 16 that is predicting the evaluation of the
Class and instructor.

107. This diagram shows that it is the expected grade
at Week 16 that is predicting the evaluation of the
Class and instructor.
The other grades are being used
by students to estimated the expected grades.