1. Public Reception of Dr. Guthrie
With General Discussion of Various Statistical Questions following the Report
By Scott Graham
Ma 405
5-5-15
Note: this is written for a technical audience (a math professor). While non-technical audiences are welcome to read this,
please understand there may be a large amount of undefined statistical vocabulary in this report.
3. 3
Abstract
Procedures
We did a study to test the public reception of a fictional soft drink, Dr. Guthrie (Dr. G). We had each
recipient rate the taste of Dr. G on a Likert Scale of 1 to 5 and then had them compare it with Dr. Pepper
(Dr. P) to see which they liked better. This will tell us how to market the soft drink.
We then analyzed the data the above results and compared them to the demographics of our
population. The specific independent demographics were Gender, Age, Minority Status, and Education
Level. We used an ANOVA test to compare the means and see exactly which demographics were the
most successful; Regression to predict the reaction of a male minority that is 39 years old with 13 years
of education and to find out which factors were significant; a contingency study to see if taste and favor
were dependent on certain factors.
Conclusions
ANOVA
We should market primarily to elderly females in the minority. They seem to receive Dr. G better than
the other demographics. The only demographic we should avoid in general is the Youth. They did not
receive Dr. G very well at all (taste=2.78 on a Likert Scale 1 to 5). The rest of the demographics received
Dr. G fairly well. We could market to them and still get some profit.
In competing with Dr. P, we should mainly worry about Age and Education Level. For Age, we will do
better in the younger age groups. For education level, we will do better with those that have a high
school education. While ANOVA shows that minority status does not make a difference in Favor, it does
show that we will succeed over 50% of the time in advertising to non-minorities with 95% confidence.
Regression
For Taste, we found that Education level is not significant in predicting a person’s enjoyment of Dr. G.
We then came up with the following model to predict their rating on a Likert Scale of 1 to 5.
.0591.229 .913 .894gen age minTaste X X X
For Favor, we found that Gender and Minority Status were not significant in predicting whether or not a
person liked Dr. G over Dr. P. We came up with the following model in which 1 means he will choose Dr.
G and 0 means he will choose Dr. P.
. .022 .655068age edFavor X X
Contingency
The only new thing Contingency told us is that at α=.05, we do not need to worry about Minority Status
as a predictor of Taste. However, the other two tests said we should pay attention to minority status so
I recommend we do pay attention to Minority Status.
4. 4
Sample Characteristics
To test the acceptance of Dr. Guthrie soft drink, we surveyed 50 people with varying ages, degrees of
education, genders, and ethnicities.
Gender
Out of 50 people, 28 were women and 22 were men.
Ethnicity
16 of the sample were in the ethnic minority
34 were non-minorities
Gender
Men
44%
Women
56%
Percent Minorities
Non-minorities
68%
Minorities
32%
5. 5
Ages
The ages for the sample were distributed as seen below:
Age Frequency
Youth (0-19) 9
Young Adult (29-39) 24
Middle Aged (40-71) 17
Average Age: 33.2
0
10
20
30
40
50
60
Percent
Age
Age Distribution
0 10 20 30 40 50 60 70 80
Age
Age Statistics Boxplot
6. 6
Education Level
The education level was only in number of years so I had to make some assumptions about the degrees
of education attained. I assumed the following for degrees of education:
Degree Years of Education Frequency in Sample
Elementary (k-6) 7 13
High School (7-12) 13 26
College/Grad 14+ 11
The education levels of the people in the sample are seen below:
0
10
20
30
40
50
60
Percent
Education Degree
Education Distribution
7. 7
0
5
10
15
20
25
30
35
40
Percent
Taste (Likert Scale)
Distribution of Likert Ratings
Series1
2%
10%
20%
38%
30%
General Survey Results
Comparison with Dr. Pepper
Overall, the participants enjoyed Dr. Guthrie more than Dr. Pepper.
Likert Rating
Overall, Dr. G was well received by the participants. With 95% confidence, the data shows that Dr. G is
liked by the general public since the mean is greater than 3.
Mean 3.84
Standard Deviation 1.04
p-value for >3 3E-7
Preferred Soda
Dr.
Pepper
36%
Dr.
Guthrie
64%
8. 8
Correlations
Taste
Collinearity of Variables
Taste on Likert
Scale Gender Age Minority Status Education
Pearson
Correlation
Taste on Likert Scale 1.000 .411 .691 .274 .073
Gender .411 1.000 -.142 -.169 -.005
Age .691 -.142 1.000 -.050 .097
Minority Status .274 -.169 -.050 1.000 .106
Education .073 -.005 .097 .106 1.000
Sig.
(1-tailed)
Taste on Likert Scale . .002 .000 .027 .307
Gender .002 . .163 .120 .486
Age .000 .163 . .364 .251
Minority Status .027 .120 .364 . .232
Education .307 .486 .251 .232 .
From this, we see that the only dependent variable at α=.05 is Taste. The other variables are
independent of each other.
Favor
Collinearity
Favor Gender Age Minority Status Education
Pearson Correlation Favor 1.000 .091 -.590 -.111 .479
Gender .091 1.000 -.142 -.169 -.005
Age -.590 -.142 1.000 -.050 .097
Minority Status -.111 -.169 -.050 1.000 .106
Education .479 -.005 .097 .106 1.000
Sig. (1-tailed) Favor . .266 .000 .222 .000
Gender .266 . .163 .120 .486
Age .000 .163 . .364 .251
Minority Status .222 .120 .364 . .232
Education .000 .486 .251 .232 .
From this, we see that the only dependent variable is Favor at α=.05. The other variables are
independent of each other.
9. 9
ANOVA
Gender
We found that on average, the females liked Dr. G more than the males. However, there was no
difference in their preferences to Dr. G and Dr. P.
Taste
Descriptives
N Mean
95% Confidence Interval for Mean
SigLower Bound Upper Bound
Male 22 3.36 2.86 3.87 .003
Female 28 4.21 3.91 4.52
Total 50 3.84 3.55 4.13
On average, the females gave Dr. G a rating .85 higher rating than the men on a Likert Scale 1 to 5.
H0: men=women
H1: men<women
p=.003<α=.05, ∴Reject H0.
From the data, we see that on average, females like Dr. G better than Dr. P.
Note: Homoscedasticity is not violated with p=.033<α-.05
Conclusion: We should market more to females than to males.
Men
Women0.0%
20.0%
40.0%
60.0%
Men
Women
Gender by Taste
10. 10
Favor
N Mean
95% Confidence Interval for Mean
SigLower Bound Upper Bound
Male 22 .59 .37 .81 .531
Female 28 .68 .49 .86
Total 50 .64 .50 .78
H0: men=women
H1 men<women
p=.531>α=.05 ∴Reject H0.
From the data, we see that on average, females like Dr. G better than Dr. P more than men do.
Note: Homoscedasticity is violated with p=.245>α=.05
Conclusion: We should not worry about gender when competing with Dr. P.
11. 11
Age
We found that enjoyment of Dr. G increases with age. However, the elderly prefer Dr. P over Dr. G more
than the middle aged and youth. Therefore, we should market mostly to the elderly unless we will be
competing directly with Dr. P.
Taste
Descriptives
N Mean
95% Confidence Interval for Mean
Lower Bound Upper Bound
Youth 9 2.78 2.14 3.42
Middle Aged 24 3.67 3.28 4.05
Elderly 17 4.65 4.34 4.96
Total 50 3.84 3.55 4.13
Scheffe Multiple Comparisons
(I) Age Grp (J) Age Grp Mean Difference (J-I) Sig.
Youth
Middle Aged
Middle Aged .889
*
.026
Elderly 1.869
*
.000
Elderly .980
*
.002
On average, the elderly gave Dr. G a rating .98 higher rating than the middle aged who gave it a
rating .89 higher than the youth on a Likert Scale of 1 to 5.
H0: Youth=Middle=Elderly
H1: Youth<Middle<Elderly
All p<α=.05, ∴Reject H0.
From the data, we see that on average, the elderly like Dr. G more than the middle aged which
likes Dr. G more than the youth.
Note: Homoscedasticity is violated with p=.337>α=.05
Conclusion: We should market mostly to the elderly, some to the middle aged, and very little to
the youth. (Graph on next page).
13. 13
0%
20%
40%
60%
80%
100%
Youth Young
Adults
Middle
Aged
Dr. P
Dr. G
Age Group by
Preference
Favor
Descriptives
N Mean
95% Confidence Interval for Mean
Lower Bound Upper Bound
Youth 9 1.00 1.00 1.00
Middle Aged 24 .75 .56 .94
Elderly 17 .29 .05 .54
Total 50 .64 .50 .78
Scheffe Multiple Comparison
(I) Age Grp (J) Age Grp
Mean Difference
(I-J) Sig.
Youth
Middle Aged
Middle Aged .250 .311
Elderly .706
*
.001
Elderly .456
*
.005
On average, the females gave Dr. G a rating .85 higher rating than the men on a Likert Scale
H0: Youth=Middle=Elderly
H1: Youth>Middle>Elderly
PYM.31>α=.05, PYE=.001<α=.05, PME=.005<α=.05, ∴Reject H0.
From the data, we see that on average, there is no provable difference in the preferences of the
youth and middle aged. Both tend to prefer Dr. G. However, there is a difference between the
preferences of the elderly and the youth and middle aged. The elderly tend to prefer Dr. P more.
Note: Homoscedasticity not is violated with p=.0+>α=.05
Conclusion: In competing with Dr. P, we should not market to the elderly but more to the youth
and middle aged.
14. 14
0%
10%
20%
30%
40%
50%
Strong
Dislike
Dislike Neutral Like
Strong
Like
Non Minorities
Minorities
Ethnicity by Taste
Minority Status
We found that, on average, minorities like Dr. Guthrie more than non-minorities at α=.06. We can
therefore market to minorities more than to non-minorities. However, it is safe to compete with Dr.
Pepper in non-minorities but not in minorities.
Taste
Descriptives
N Mean
95% Confidence Interval for Mean
Sig.Lower Bound Upper Bound
Non-Minority 34 3.65 3.27 4.02 .054
Minority 16 4.25 3.79 4.71
Total 50 3.84 3.55 4.13
On average, the minorities gave Dr. G a rating .6 higher rating than the non-minorities on a Likert
Scale 1 to 5.
H0: minorities=non-minorities
H1: minorities>non-minorities
p=.054>α=.05, ∴Do not reject H0.
From the data, we see that on average there is not much of a provable difference between
minorities and non-minorities in their enjoyment of Dr. G.
Note: Homoscedasticity is violated with p=.185>α=.05
Conclusion: We cannot reject H0 at α=.05. However, we can reject it at α=.06. That is still plenty
so we can still fairly safely market to minorities more than non-minorities.
15. 15
Favor
Descriptives
N Mean
95% Confidence Interval for Mean
SigLower Bound Upper Bound
Non-Minority 34 .68 .51 .84 .444
Minority 16 .56 .29 .84
Total 50 .64 .50 .78
On average, the minorities gave Dr. G a rating .14 higher rating than the non-minorities on a Likert
Scale 1 to 5.
H0: minorities=non-minorities
H1: minorities<non-minorities
p=.444>α=.05, ∴Do not reject H0.
While there is no provable difference from the data in preference between minorities and non-
minorities, we can say with 95% confidence that most non-minorities prefer Dr. G while we cannot
say the same about minorities.
Note: Homoscedasticity is violated with p=.217>α=.05
Conclusion: Avoid competing with Dr. Pepper for minorities but do not avoid non-minorities.
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Non-minority Minority
Dr. P
Dr. G
Ethnicity by Preference
16. 16
0.0%
50.0%
Strong…
Dislike
Neutral
Like
StrongLike
Elementary
Secondary
Post-Secondary
Education by
Education
According to the data, education level does not affect a person’s enjoyment of Dr. G. However, we
found that those with a high school education tended to prefer Dr. G over Dr. P more than those with a
Gradeschool or College education.
Taste
Descriptives
N Mean Std. Error
95% Confidence Interval for Mean
Lower Bound Upper Bound
Gradeschool 16 3.81 .306 3.16 4.46
High School 17 3.88 .241 3.37 4.39
College/Grad 17 3.82 .231 3.33 4.31
Total 50 3.84 .147 3.55 4.13
Sheffe Multiple Comparisons
(I) Ed Grp (J) Ed Grp
Mean Difference
(J-I) Sig.
95% Confidence Interval
Lower Bound Upper Bound
Gradeschool
High School
High School .070 .982 -1.00 .86
College/Grad .011 1.000 -.94 .92
College/Grad -.059 .987 -.86 .98
On average, the college/grad people gave Dr. G a rating .01 lower rating than the gradeschool
people who gave it a rating .05 lower than the high school on a Likert Scale of 1 to 5.
H0: Grades=High=College
H1: Grades <College<High
All p>α=.05, ∴Do not reject H0.
From the data, we see that on average,
education level makes no provable difference
a person’s enjoyment of Dr. G.
Note: Homoscedasticity is violated with
p=.434>α=.05
Conclusion: We should not look at education
level to find out who will enjoy Dr. G.
17. 17
Favor
Descriptives
N Mean Std. Error
95% Confidence Interval for Mean
Lower Bound Upper Bound
Gradeschool 16 .38 .125 .11 .64
High School 17 .53 .125 .26 .79
College/Grad 17 1.00 .000 1.00 1.00
Total 50 .64 .069 .50 .78
Scheffe Multiple Comparison
Multiple
Comparisons
On average, the females gave Dr. G a rating .85 higher rating than the men on a Likert Scale
H0: Grades=High =College
H1: Grades <High<College
PGH=.57>α=.05, PGC=0+<α=.05, PHC=.007<α=.05, ∴Reject H0.
From the data, we see that on average, there is no provable difference in the preferences of the
gradeschool and high school groups. However, there is a difference between the preferences of
the college and the gradeschool and high school. The college group tends to prefer Dr. G. more.
Note: Homoscedasticity is not violated with p=.0+>α=.05
Conclusion: In competing with Dr. P, our time would be better spent advertising to those with a
college/grad level education. Advertising to lower education groups would be very risky.
(I) Ed Grp (J) Ed Grp
Mean Difference
(J-I) Sig.
95% Confidence Interval
Lower Bound Upper Bound
Gradeschool
High School
High School .154 .565 -.52 .21
College/Grad .625
*
.000 -.99 -.26
College/Grad .471
*
.007 -.83 -.11
18. 18
Regression
From the data, we found that, in analyzing taste, Gender, Age, and Minority Status were all
significant factors in predicting a person’s enjoyment of Dr. G. Education, however, was not a
significant factor.
Also from the data, we found that only Age and Education were significant factors in a person’s
preference between Dr. P and Dr. G. Gender and Minority Status were not significant factors in
predicting a person’s favor.
Taste
Multiple Linear Regression
Coefficients with Significance
Model Variable
Unstandardized
Coefficients
B Sig.
(Constant)
Gender (Xgen)
Age(Xage)
Minority Status(Xmin)
Education(Xed)
1.009
1.232
.060
.924
-.012
.000
Βgen .000
Βage .000
Βmin .000
Βed .312
From the data, we see that at α=.05, all of the above factors excluding education (pEd=.312>α) are
significant in the prediction of a test statistic’s taste
We believe we can to some degree predict how much a person will like Dr. G with the following
equation:
1.232 .012 1.0. 0906 .924
gen gen age age min min ed ed
gen age min ed
Taste X X X X k
X X X X
Example: a male minority that is 39 years old and has 13 years of education will, on average, give
Dr. Guthrie a rating of 4.12 on a Likert Scale 1 to 5.
Note: from the table of correlations, all of the variables above besides Taste are independent of
each other.
19. 19
Stepwise Regression
R-Squares of Model
Model
R
Square
Adjusted R
Square
Age .477 .466
Age, Gender .742 .731
Age, Gender, Minority Status .908 .902
The Stepwise Regression did not include Education in its model. Therefore, we can eliminate that
from our model giving us:
gen gen age age min minTaste X X X k
Therefore, do not analyze education level in determining whether or not to advertise to them.
Here is another Regression model calculated without Minority:
Regression without Education
Model
Unstandardized
Coefficients
B Sig.
(Constant)
Gender
Age
Minority Status
.894 .000
1.229 .000
.059 .000
.913 .000
Here is the model we came up without Education.
.0591.229 .913 .894gen age minTaste X X X
Example: a male minority that is 39 years old and has 13 years of education will, on average, give
Dr. Guthrie a rating of 4.108 on a Likert Scale 1 to 5.
20. 20
Favor
Multiple Linear Regression
Coefficients with Significance
Model
Unstandardized
Coefficients
B Sig.
(Constant)
Gender
Age
Minority Status
Education
.732 .000
-.035 .682
-.023 .000
-.216 .019
.071 .000
From the data, we see that at α=.05, all of the above factors excluding Gender (pgen=.682>α) and
Minority Status (pmin=.222>α) are significant in the prediction of a test statistic’s taste
We believe we can to some degree predict how much a person will like Dr. G with the following
equation:
.035 .023 –.0216 .071 .732
gen gen age age min min ed ed
gen age min ed
Favor X X X X k
X X X X
Example: a male minority that is 39 years old and has 13 years of education will, on average, favor
Dr. G 54.2% of the time.
Note: from the table of correlations, all of the variables above besides Taste are independent of
each other.
21. 21
Stepwise Regression
R-Squares of Model
Model R
R
Square
Age .590
a
.348
Age, Education .799
b
.639
Age, Education, Minority Status .824
c
.679
The Stepwise Regression did not include Education in its model. Also, eliminating Minority Status
only changes the R-Square by .04. Therefore, we can eliminate those from our model giving us:
age age ed edFavor X X k
Regression without Gender and Minority Status
Model
Unstandardize
d Coefficients
B Sig.
(Constant)
Age
Education
.655 .000
-.022 .000
.068 .000
Here is our new model for predicting Favor:
. .022 .655068age edFavor X X
Example: a male minority that is 39 years old and has 13 years of education will, on average,
prefer Dr. G 68.1% of the time.
22. 22
Contingency
Taste
Pearson
Chi-Square
Phi Sig
Gender 11.30 .475 .023
Age 39.25 .888 .000
Education 4.413 .297 .818
Minority Status 4.713 .307 .318
According to this analysis, there is no significant difference in enjoyment of Dr. G between
difference in Education levels and Minority Status (α=.05). We should only study Age and Gender.
Favor
Pearson
Chi-Square
Phi Sig
Gender .411 .091 .522
Age 15.150 .550 .001
Education 15.342 .554 .000
Minority Status .613 .111 .434
According to this analysis, there is no significant difference in enjoyment of Dr. G between
differences in Gender or Minority Status (α=.05). We should only study Age and Education.
23. 23
General Discussion
I. Discuss the problem with the type I error in repeated t-test and how to
handle that problem by
A. Bonferroni’s correction on Alpha
In hypothesis testing, pi=α is the probability that a Type I error will occur for a single
test, i. When n hypothesis tests are used, the probability of one Type 1 error is
approximately ip n . Therefore, to keep the actual p to a minimum, we must
divide α by n to get the correct p=α. For example, if we want 95% overall confidence for
five testes, we would have to use α=.01 for each individual test. This will give as an
overall α=.05.1
B. Multiple test as ANOVA and Regression.
ANOVA and Regression don’t need to be adjusted because they are, by definition, all
under the same α.2
II. For ANOVA
A. Discuss the type of variables (dependent = performance and independent =
influencing)
In ANOVA, we study whether or not the performance of a variable is influenced by the
characteristics on another variable. For example, a person’s performance in grades may
be influenced by his gender. Another way to say this is his grades are dependent (to a
degree) on his gender. His gender is not dependent of any other variables in the test. It
is independent.3
B. Discuss the assumptions to do an ANOVA test.
1. Random Sample: if it is not a random sample, we could prove anything.
2. Dependent variable is normally distributed: The equation for an ANOVA uses a Normal
distribution.
3. Population Variances must be equal. This is required for the formula to work. Otherwise it
is very complicated.4
C. How would you handle ANOVA data that violated homoscedasticity?
First, make sure the violation is significant enough to worry about. Many times,
heteroscedasticity occurs because one mean is much larger than the other. Try using a
log transformation. Also, you can try to use a Welch’s ANOVA.5
1
Guthrie, Gary. "Intro for Final Project." Ma 405. United States, Greenville. 22 Apr. 2015. Lecture.
2
Abbey Young (paraphrasing Gary Guthrie)
3
Guthrie, Gary.
4
"Hypotheses Statements and Assumptions for One−Way ANOVA." Hypotheses Statements and Assumptions for
One−Way ANOVA. N.p., n.d. Web. 04 May 2015.
24. 24
D. What is tested under ANOVA?
ANOVA tests the differences of means to see if they are significant.6
They are also
useful because they give us a measure of confidence as well.
E. How does one use the results of ANOVA to do confidence intervals for the average
score in a group? If one graphed all of the confidence intervals what would we like to
see?
Instead of using the t-distribution, we can use the F-distribution to find the critical
values of a sample.7
If we graphed the confidence intervals for multiple tests on the
same subject with unchanging factors, we would ideally like to see all of the confidence
intervals placed around the same area.
F. Discuss how to do ANOVA with regression
Since the coefficients in regression are the differences in means in ANOVA from a
reference variable, the p-values in regression give us the significance of the differences
in the means from the reference variable. The two are pretty much the same thing.
One analyzes the means, the other analyzes the difference in means.8
III.For regression
A. Discuss the type of variables (dependent = performance and independent =
influencing)
This is the same as in ANOVA. The dependent variables are influenced by the
independent variables. The purpose of regression is to measure how much the
influence of the independent variables affects the performance of the dependent
variables.9
B. Discuss the assumptions to do a regression analysis.
1. For Linear, it there must be a linear relationship. The data sample points must form a
somewhat straight line. Same for Quadratic, Log, etc.
2. Multivariate Normality
3. Little or no multicolinearity: the independent variables may not influence each other in the
test.
4. No autocorrelation: residuals may not be independent from each other.
5. Homoscedasticity: the variances must be equal.10
5
McDonald, John H. "Handbook of Biological Statistics." Homoscedasticity -. N.p., 4 Dec. 2014. Web. 04 May 2015.
6
"Analysis of Variance (ANOVA)." Analysis of Variance (ANOVA). N.p., n.d. Web. 04 May 2015.
7
"Confidence Interval for ANOVA." Real Statistics Using Excel. N.p., n.d. Web. 05 May 2015.
8
Grace-Martin, Karen. "Why ANOVA and Linear Regression Are the Same Analysis." The Analysis Factor RSS. N.p.,
n.d. Web. 05 May 2015.
9
Guthrie, Gary
10
"Assumptions of Linear Regression - Statistics Solutions." Statistics Solutions. N.p., n.d. Web. 05 May 2015.
25. 25
C. Discuss how each of the following can be used to analyze how good a regression
model fits
1. R2
R2
tells us how much of our dependent variable can be predicted by our independent
variables, on average. If R2
is .1, our model is predicting 10% of the dependent
variable.11
2. Residuals and sum of residuals squared
The residuals are simply individual differences between the observed data points and
the expected data points from the model. These typically cannot tell us very much as
some will be negative and some will be positive so their sum can be zero even if it is a
horrible fit. The sum of residuals squared, however will always be positive. It will tell us
how much residual we have in our prediction.12
3. 0 1: ... 0nH
We multiply the coefficients βi by the characteristics of or surrounding the test subjects.
If the coefficients are smaller, they will have less effect on the dependent variable. If we
cannot reject H0, the model is useless because the coefficients cannot predict the
dependent variable.13
D. Discuss how one does a parsimony study.
A parsimony goes through the different factors and analyzes them to see which ones are
useful in analyzing data and in constructing a model. One technique is using a stepwise
regression and seeing how much R2
each of the independent variables contains. If they
contain less than .05, we can disregard that factor as insignificant in our study. After
finding the useless factors, go through and do another regression study excluding the
rejected factors.
E. How does one use the results of regression to do confidence intervals for an
individual’s score that has certain demographic characteristics?
Regression Analysis in SPSS returns the standard error along with the confidence
interval. In this test, we want to see if any of the coefficients equal zero. If zero is in the
confidence interval, we can reject that demographic since it is insignificant.
IV. For contingency – give the questions and answers.
A. What is a contingency table?
A contingency table (also known as a Cross Tab table) looks at the divisions within
groups of different factors and looks at how they are distributed across the dependent
discrete results. It can be done in counts, percent, or both. Note: if dealing with
11
"Goodness-of-Fit Statistics." Goodness of Fit Statistics. N.p., n.d. Web. 05 May 2015.
12
Guthrie, Gary
13
Guthrie, Gary
26. 26
continuous data, break up the data into several ranges.14
For two variables Ai and Bi, a
2x2 contingency table looks like the following:
B1 B2 Total
A1
1 1A B 1 2A B 1An
A2
2 1A B 2 2A B 2An
Total
1Bn 2Bn 1 2 1 2A A B Bn n n n
B. Discuss the type of variables (dependent = performance and independent =
influencing)
The goal is to see whether or not the independent variables influence the dependent
variables. It does this by grouping the results into different categories. We can then
look at the resulting table to see if it looks like the variables are dependent and use it in
the calculation of the Chi-Square
C. Discuss Pearson’s Chi-Square.
The Pearson Chi-Square test is used to see if the variables are independent of each
other. It starts with the null hypothesis that the two variables are independent of each
other. Two variables A and B in the above table are independent if
1 2 1 2
i j
i j
A A B B
A B
A B
n n n n
.
This is t called the expected value. The Chi-Square finds the sum of the squares of the
differences between each factor’s expected and observed values divided by the
expected value. The higher the Chi-Square, the more likely the variables are
independent. We then use the p-value for hypothesis testing to test the null
hypothesis.15
D. Explain Phi and Cramer’s V
Phi is simply a square root of a Chi-Squared divided by n. It is bounded by 0 and 1.0 and
tells us how predictable one variable is given another variable. Note: Phi is only good
for 2x2 contingency tables.
Cramer’s V is just like Phi except that it also divides by the smaller of (rows-1) or
(columns-1). Therefore, for a 2x2, Cramer’s V=Phi. However, it is useful because it will
not go above 1.0 for tables larger than 2X2. It tells us the same as Cramer’s V.16
14
Stockburger, David W. "Chi-Square and Tests of Contingency Tables." Chi-Square and Tests of Contingency
Tables. N.p., n.d. Web. 05 May 2015.
15
"Pearson's Chi-Square Test for Independence." Upenn. N.p., n.d. Web. 05 May 2015.
16
"Nominal Association: Phi and Cramer's V." Measures of Nominal Level Association. N.p., n.d. Web. 05 May
2015.
27. 27
Bibliography
"Analysis of Variance (ANOVA)." Analysis of Variance (ANOVA). N.p., n.d. Web. 04 May 2015.
"Assumptions of Linear Regression - Statistics Solutions." Statistics Solutions. N.p., n.d. Web. 05 May
2015.
"Confidence Interval for ANOVA." Real Statistics Using Excel. N.p., n.d. Web. 05 May 2015.
"Goodness-of-Fit Statistics." Goodness of Fit Statistics. N.p., n.d. Web. 05 May 2015.
Grace-Martin, Karen. "Why ANOVA and Linear Regression Are the Same Analysis." The Analysis Factor
RSS. N.p., n.d. Web. 05 May 2015.
Guthrie, Gary. "Intro for Final Project." Ma 405. United States, Greenville. 22 Apr. 2015. Lecture.
"Hypotheses Statements and Assumptions for One−Way ANOVA." Hypotheses Statements and
Assumptions for One−Way ANOVA. N.p., n.d. Web. 04 May 2015.
McDonald, John H. "Handbook of Biological Statistics." Homoscedasticity -. N.p., 4 Dec. 2014. Web. 04
May 2015.
"Nominal Association: Phi and Cramer's V." Measures of Nominal Level Association. N.p., n.d. Web. 05
May 2015.
"Pearson's Chi-Square Test for Independence." Upenn. N.p., n.d. Web. 05 May 2015.
Stockburger, David W. "Chi-Square and Tests of Contingency Tables." Chi-Square and Tests of
Contingency Tables. N.p., n.d. Web. 05 May 2015.
"Welcome to the Institute for Digital Research and Education." Repeated Measures Anova. N.p., 31 June
1997. Web. 04 May 2015.
