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## 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
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