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Table Of Contents
  • Computing Central Tendency, Variance and Standard Deviation
  • Paired Sample T-test
  • Two-sample T-test Analysis
  • ANOVA F-statistics

Computing Central Tendency, Variance and Standard Deviation

The first data set includes two tables representing two classes of a college course.
Calculate and report the following for the quiz, midterm exam, final exam, and total grade for class one – mean, median, mode, range, variance, and standard deviation.
Quiz   Midterm Exam   Final Exam   Total Grade  
Mean                  7.55  Mean                 77.40  Mean                 78.20  Mean                 81.72
Median                  7.00  Median                 77.00  Median                 76.00  Median                 80.64
Mode                  7.00  Mode                 77.00  Mode                 73.00  Mode #N/A
Standard Deviation                  1.36  Standard Deviation                 11.39  Standard Deviation                 11.02  Standard Deviation                   8.66
Sample Variance                  1.84  Sample Variance               129.62  Sample Variance               121.43  Sample Variance                 75.02
Range                  5.00  Range                 37.00  Range                 37.00  Range                 29.11

Paired Sample T-test

Determine if there is a meaningful difference between midterm exam scores and final exam scores for the students in class one.

Paired sample t-test for the midterm and final score show that the difference between the scores are not statistically significant as the p-value is 0.52 which is greater than level of significance.

t-Test: Paired Two Sample for Means
  Midterm Exam Final Exam
Mean                     77.40               78.20
Variance                   129.62             121.43
Observations                     20.00               20.00
Hypothesized Mean Difference                            -
df                     19.00
t Stat                     (0.65)
P(T<=t) two-tail                        0.52
t Critical two-tail                        2.09  

Two-sample T-test Analysis

Determine if there is a meaningful difference between the average midterm exam scores for class one and class two. Determine if there is a meaningful difference between the average final exam scores for class one and class two.
The two-sample t-test indicates that the difference between averages of two midterm exam scores are not statistically significant.
t-Test: Two-Sample Assuming Equal Variances
  Midterm Exam Midterm Exam
Mean                     77.40               74.80
Variance                   129.62             114.37
Observations                     20.00               30.00
Pooled Variance                   120.41
Hypothesized Mean Difference                            -
df                     48.00
t Stat                        0.82
P(T<=t) two-tail                        0.42
t Critical two-tail                        2.01  

The two-sample t-test indicates that the difference between averages of two midterm exam scores are not statistically significant as p-value 0.22 is greater than 0.05.

t-Test: Two-Sample Assuming Equal Variances
  Final Exam Final Exam
Mean                     78.20               74.30
Variance                   121.43             119.32
Observations                     20.00               30.00
Pooled Variance                   120.16
Hypothesized Mean Difference                            -
df                     48.00
t Stat                        1.23
P(T<=t) two-tail                        0.22
t Critical two-tail                        2.01  

ANOVA F-statistics

Using the scores for both class one and class two together (treating them collectively as a cohort), determine if midterm exam score is a significant predictor of final exam score. In roughly what percent of cases would knowing someone’s midterm exam score allow us to predict their final exam score?
Yes. The midterm score can be used to predict the final score of students with pretty high confidence. The mid term score explains around 69% of variance in final score and the ANOVA F-statistic yields statistically significant result (F = 108.64, p <0.001) indicating that the midterm score is statistically significant in predicting the final exam score.
ANOVA
  df SS MS F Significance F
Regression                  1.00               4,126.72         4,126.72             108.64                     0.00
Residual               48.00               1,823.30               37.99
Total               49.00               5,950.02      
 
  Coefficients Standard Error t Stat P-value
Intercept               12.22                        6.17                  1.98                  0.05
Midterm Exam                  0.84                        0.08               10.42                  0.00

The regression model also gives an estimate of coefficient which means for each unit increase in midterm score, average increase in final exam score is 0.84.

Bonus: Using only the scores from class one, and looking at the three variables quiz score, midterm exam score, and final exam score, which of these variables are significant predictors of total grade in the course?

Yes. The quiz score, midterm score, final exam score is highly efficient in predicting the total score with R2 of 0.91 indicating that regression model was able to explain 91% of variance in the data. The ANOVA F-statistic also yields significant result with F = 52.07, p<0.001.

ANOVA
   df  SS  MS  F  Significance F
Regression                  3.00               1,292.87             430.96               52.07                     0.00
Residual               16.00                   132.42                  8.28
Total               19.00               1,425.29      

   Coefficients  Standard Error  t Stat  P-value
Intercept               22.66                        4.82                  4.70                  0.00
Quiz                  0.01                        0.73                  0.02                  0.99
Midterm Exam                  0.39                        0.14                  2.76                  0.01
Final Exam                  0.37                        0.13                  2.94                  0.01

Among the three predictors, midterm score and final exam score were statistically significant in predicting the total grade, but the quiz score was not!

The second data set includes one table providing data on students in four college majors, including scores on an anxiety inventory, GPA in major, and GPA overall.

What are the frequencies of each major?In looking at the frequency of each major, do you observe any potential issue? How do you think that issue might be resolved?

Major Count of Major
Business 8
Philosophy 6
Pre-Med 14
Psychology 12
Grand Total 40
There are total of four major – Business (8), Philosophy (6), Pre-Med (14), and Psychology (12). The issue here is that the number of samples for Business and Philosophy is quite small as compared to Pre-Med and Psychology. It can be resolved by either getting more data or resampling the data from respective samples.

Which major has the highest average measured anxiety? Which major has the lowest average measured anxiety?Comparing each of the four majors against each other at once, is there a meaningful difference in the average anxiety score?
Row Labels Average of Anxiety
Business                              17.00
Philosophy                              15.00
Pre-Med                              18.93
Psychology                              17.00

Pre-Med has highest level of anxiety and Philosophy has lowest level. The ANOVA test shows that there is not significant difference across groups in Anxiety levels
ANOVA
Source of Variation  SS  df  MS  F  P-value  F crit
Between Groups                70.45        3.00                23.48                  1.18                  0.33                  2.87
Within Groups             718.93      36.00                19.97
Total             789.38      39.00        

Comparing each of the four majors against each other at once, is there a meaningful difference in the average GPA in major? Comparing each of the four majors against each other at once, is there a meaningful difference in the average overall GPA?

No. There is no significant differences in Average GPA in majors across different majors.

ANOVA
Source of Variation  SS  df  MS  F  P-value  F crit
Between Groups                  0.53      3.00                  0.18                  1.42                  0.25                  2.87
Within Groups                  4.45    36.00                  0.12
Total                  4.98    39.00        
The ANOVA F-test shows statistically insignificant result with F = 1.42, p = 0.25.

For the entire group of students, is there a meaningful difference between GPA in major and overall GPA? For the entire group of students, is anxiety a significant predictor of overall GPA?
Based on paired sample t-test, there is significant difference between GPA in major and overall GPA with t = 4.47, p<0.001.
t-Test: Paired Two Sample for Means
  GPA in Major Overall GPA
Mean                    3.48                  3.33
Variance                    0.13                  0.15
Observations                  40.00               40.00
Pearson Correlation                    0.83
Hypothesized Mean Difference                         -
df                  39.00
t Stat                    4.47
P(T<=t) two-tail                    0.00
t Critical two-tail                    2.02  

We also use regression model to test if Anxiety is significant predictor of overall GPA or not. It turns out that Anxiety level is significant predictor of overall GPA with F = 7.74, p = 0.01. The model has multiple R2 = 0.17 indicating that Anxiety account for 17% of variance in the overall GPA.

ANOVA
   df  SS  MS  F  Significance F
Regression                    1.00                  0.97           0.97           7.74           0.01
Residual                  38.00                  4.75           0.13
Total                  39.00                  5.72      
   Coefficients  Standard Error  t Stat  P-value
Intercept                    2.72                  0.23         12.05           0.00
Anxiety                    0.04                  0.01           2.78           0.01

The estimated coefficients show that the anxiety has significant positive relationship with the GPA. Each unit increase in Anxiety, on an average, increases overall GPA by 0.04.

Bonus: For the entire group of students, determine the magnitude and direction of the relationship between grade-point-average in major and grand-point-average overall. You should have been able to guess what you would find before calculating anything or even looking at a single value; why?

Based on my guess, I would say the GPA in Major would be highly correlated to the overall GPA and the relationship will be positive. This is because we expect student who score high for major, to score for other courses as well. Similarly, for students who are unable to score in majors, it is expected that they would score low in other courses as well.

The guess is verified by the data:

Regression Statistics
Multiple R                  0.83
R Square                  0.69
Adjusted R Square                  0.68
Standard Error                  0.22
Observations 40
The GPA in major is positively correlated with overall GPA.
Bonus: Is there another effect you think would be interesting to explore with these data? Why?
We can use Anxiety as control variable and then see if different majors have significantly different GPA after controlling for Anxiety levels.
Data Set 2 – Four Majors
Participant Anxiety GPA in Major Overall GPA Major
1 17 3.60 3.36 Pre-Med
2 12 3.33 3.00 Psychology
3 13 3.67 3.38 Philosophy
4 16 4.00 3.70 Pre-Med
5 13 3.50 3.30 Psychology
6 17 3.33 3.13 Business
7 13 3.50 3.45 Philosophy
8 21 3.80 3.75 Pre-Med
9 15 3.00 3.00 Pre-Med
10 23 2.50 2.70 Business
11 17 3.75 3.18 Psychology
12 24 3.80 3.89 Psychology
13 24 4.00 3.80 Philosophy
14 22 3.75 3.40 Psychology
15 18 3.67 3.60 Pre-Med
16 14 3.67 3.56 Psychology
17 23 3.50 3.50 Business
18 13 3.67 3.00 Business
19 17 3.60 3.36 Pre-Med
20 12 3.33 3.00 Business
21 22 3.50 3.18 Pre-Med
22 19 3.50 3.10 Psychology
23 23 3.00 3.00 Pre-Med
24 25 3.25 3.18 Pre-Med
25 23 3.33 3.09 Psychology
26 24 3.67 3.80 Psychology
27 13 2.75 2.40 Business
28 11 3.00 2.81 Psychology
29 16 3.60 3.36 Pre-Med
30 14 3.00 3.30 Business
31 8 3.00 2.77 Philosophy
32 19 3.80 3.89 Pre-Med
33 13 3.00 2.67 Philosophy
34 13 3.67 3.56 Psychology
35 21 4.00 3.89 Business
36 19 3.67 3.30 Philosophy
37 18 3.67 3.80 Pre-Med
38 21 4.00 4.00 Pre-Med
39 12 3.67 3.40 Psychology
40 17 3.25 3.60 Pre-Med