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.

A

B

C

D

F

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 chi-square 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.

A

B

C

D

F

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

Observed frequencies

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).

A

B

C

D

F

Totals

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

Expected frequencies

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.

A

B

C

D

F

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

Observed frequencies - expected frequencies

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 chi2 statistic is obtained by adding up the squares of theses differences divided by the expected value in each cell.

A

B

C

D

F

Totals

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

chi2 = 728.97

The number of degrees of freedom is (4-1)(5-1) = 12. This value of is way off the charts of the critical values of chi2 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 chi-square score off to the nearest tenth. They also display the P-value which is so small that we would reject the null hypothesis.