2. A study was conducted to see if a student's year in school had any effect on how well they did in their classes. The following data for all the students in all their classes was obtained from the Registrar.






Fresh 
1569 
1567 
1214 
432 
987 
Soph 
1484 
1581 
1322 
212 
643 
Jun 
1592 
1609 
1472 
153 
421 
Sen 
1685 
1623 
1211 
75 
315 
Use a chisquare test to determine if there is a significant difference between the grades received by students in different classes.
First we need to get our row and column totals.







Fresh 
1569 
1567 
1214 
432 
987 
5769 
Soph 
1484 
1581 
1322 
212 
643 
5242 
Jun 
1592 
1609 
1472 
153 
421 
5247 
Sen 
1685 
1623 
1211 
75 
315 
4909 
Totals 
6330 
6380 
5219 
872 
2366 
21167 
Now we can enter the expected frequencies into each cell in the table. The expected frequency is computed by (row total)x(column total)/(grand total).







Fresh 
1725.22 
1738.85 
1422.42 
237.66 
644.85 
5796 
Soph 
1567.62 
1580.00 
1292.48 
215.95 
585.94 
5242 
Jun 
1569.12 
1581.51 
1293.72 
216.16 
586.50 
5247 
Sen 
1468.04 
1479.63 
1210.38 
202.23 
548.72 
4909 
Totals 
6330 
6380 
5219 
872 
2366 
21167 
Since these values have all been rounded off to the nearest hundredth, you may find slight discrepancies in the row and column totals due to round off error. We now compute the difference between the observed and expected frequencies in each cell.






Fresh 
156.22 
171.85 
208.42 
194.34 
342.15 
Soph 
83.62 
1.00 
29.52 
3.95 
57.06 
Jun 
22.88 
27.49 
178.28 
63.16 
165.50 
Sen 
216.96 
143.37 
0.62 
127.23 
233.72 
The row and column totals should all be 0 at this point. Again, in some cells you will find slight discrepancies due to round off error.
The chi^{2} statistic is obtained by adding up the squares of theses differences divided by the expected value in each cell.







Fresh 
14.15 
16.98 
30.54 
158.92 
181.54 

Soph 
4.46 
0.00 
0.67 
0.07 
5.56 

Jun 
0.33 
0.48 
24.57 
18.45 
46.70 

Sen 
32.06 
13.89 
0.00 
80.05 
99.55 

Totals 





728.97 
The number of degrees of freedom is (41)(51) = 12. This value of is way off the charts of the critical values of chi^{2} so we reject the null hypothesis. That means that students in different classes can expect to get different grades. Looking at the tables, we would conclude that the upper division students get significantly better grades than the lower division students.
The Data Desk printout looks like
you can see that they rounded the chisquare score off to the nearest tenth. They also display the Pvalue which is so small that we would reject the null hypothesis.