
Katy, Sajjad, Lee - 5/6/2010 - Stats II

We built a very short survey and collected answers via an online
questionnaire and handing it out at the Academic Support Center. The
online survey can be seen at

http://www.kwiksurveys.com/online-survey.php?surveyID=KHJJKO_c0943204

Each question was followed by 5 different choices: "Strongly Agree,"
"Agree," "Neutral," "Disagree," "Strongly Disagree."

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+                                                                             +
+   1. I choose a course based on feedback from a friend's answer to my       +
+   question: Who's the best teacher for this course?                         +
+                                                                             +
+   2. I choose a course by going online to a site which rates the professor  +
+   of the course.                                                            +
+                                                                             +
+   3. I choose a course based on my own interests and needs.                 +
+                                                                             +
+   4. I choose a course based on how easy I think it will be to get a good   +
+   grade in it.                                                              +
+                                                                             +
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

The answers were entered into Minitab. We turned "Strongly Agree" into a
5, "Agree" into a 4 ... "Strongly Disagree" into a 1. We had 24
respondents.

question 1 question 2 question 3 question 4

c5         c6         c7         c8

5          3          5          3
5          3          5          1
5          3          5          1
5          3          1          1
3          2          5          2
3          4          5          3
5          4          4          4
3          3          5          2
5          3          5          3
4          3          5          2
5          1          5          2
1          3          5          1
1          1          5          1
5          4          5          2
3          2          5          2
5          4          4          2
5          2          3          1
4          3          5          4
4          2          5          3
5          5          5          1
5          2          4          2
4          4          5          4
5          2          4          2
4          3          5          2

The hypotheses were:

H0: median scores are equal for all 4 questions
H1: at least one median score is not equal to another

The questionnaire is admittedly minimal and it is not surprising that
the alternative hypothesis won out.

One-way ANOVA: C5, C6, C7, C8

Source  DF      SS     MS      F      P
Factor   3   91.78  30.59  28.23  0.000
Error   92   99.71   1.08
Total   95  191.49

S = 1.041   R-Sq = 47.93%   R-Sq(adj) = 46.23%

                         Individual 95% CIs For Mean Based on
                         Pooled StDev
Level   N   Mean  StDev  ---+---------+---------+---------+------
C5     24  4.125  1.227                      (---*---)
C6     24  2.875  0.992          (---*---)
C7     24  4.583  0.929                           (---*---)
C8     24  2.125  0.992  (---*---)
                         ---+---------+---------+---------+------
                          2.0       3.0       4.0       5.0

Pooled StDev = 1.041

page 789 Table H gives a C.V. = 3.03

Since 28.23 > 3.03 we reject null hypothesis (big time!)

We can summarize the results in plain English in this way:

Picking a course because it's easy was rated lowest.
Using an online professor rating site was next.
Feedback from friends was next.
Personal interest and need was rated highest.

no Scheffé Tests were done here. It appears that all medians are
significantly different.


We also did a simulation of the questionnaire.

We generated 20 samples of answers. We "rigged it" so that question 1
got a "Strongly Agree" rating 75% of the time and 1 of the other 4
ratings 6.25% of the time. Questions 2, 3 and 4 got 1 of the 5 ratings
20% of the time.

p(X=5) = 3/4  p(X<5)=1/4 for question 1.
p(X=i) = 1/5 for questions 2, 3, and 4.

calc / random data / discrete - use c1 and c2 - gen. 20 rows - put in c5 : question 1
calc / random data / discrete - use c3 and c4 - gen. 20 rows - put in c6 : question 2
calc / random data / discrete - use c3 and c4 - gen. 20 rows - put in c7 : question 3
calc / random data / discrete - use c3 and c4 - gen. 20 rows - put in c8 : question 4

page 789 Table H gives a C.V. = 3.1

One-way ANOVA: s1-q1, s1-q2, s1-q3, s1-q4

Source  DF      SS    MS     F      P
Factor   3   24.34  8.11  3.58  0.018
Error   76  172.15  2.27
Total   79  196.49

S = 1.505   R-Sq = 12.39%   R-Sq(adj) = 8.93%

                         Individual 95% CIs For Mean Based on
                         Pooled StDev
Level   N   Mean  StDev    +---------+---------+---------+---------
s1-q1  20  4.150  1.461                        (--------*---------)
s1-q2  20  2.800  1.704    (---------*---------)
s1-q3  20  2.800  1.542    (---------*---------)
s1-q4  20  3.200  1.281          (---------*--------)
                           +---------+---------+---------+---------
                         2.10      2.80      3.50      4.20

Pooled StDev = 1.505

Since 3.58 > 3.1 we reject null hypothesis (just barely).

Scheffé Test

F' = (3-1)*3.1 = 6.2

19/76(1.461^2 + 1.704^2 + 1.542^2 + 1.281^2) = 2.2642155 (this is "s^2 within")
2.2642155*(1/20+1/20) = 2.2642155*(1/10) = 0.22642155

question 1 version question 2

Fs = ((x1-x2)^2)/0.22642155 = (4.150-2.800)^2/0.22642155 =
1.8225/0.22642155 = 8.04914549874

Since 8.04914549874 > 6.2 we conclude the first question's median value
is significantly different than the second question's median value
(which is expected since we rigged it that way).

Based on how the data was generated we would expect Scheffé tests for
all other combinations would show no significant difference between
questions 2, 3, and 4 and a significant difference between 1 and 3 and
between 1 and 4.

Here's the simulated data:

c1        c2        c3          c4
values-1  p-1       values-2    p-2

1         0.0625    1           0.2
2         0.0625    2           0.2
3         0.0625    3           0.2
4         0.0625    4           0.2
5         0.7500    5           0.2

c5            c6            c7            c8
question 1    question 2    question 3    question 4

5             5             4             3
5             2             1             3
4             5             5             2
5             4             3             4
5             1             4             3
5             2             1             4
5             2             3             5
5             5             2             1
5             1             3             3
5             5             1             5
5             4             5             2
5             1             5             3
1             1             2             5
5             2             1             2
2             1             1             4
5             5             2             5
2             4             5             2
5             1             1             3
1             1             4             1
3             4             3             4

