Question

Blood tests were performed on three groups of patients that had cornoavirus and suffered mild, severe or critical symptoms.

The immune system makes antibodies to protect the body from viruses and the levels of the IgG antibody was measured in each patient.  IgG is the most common antibody in blood and the researchers wanted to examine if there were different levels within the three groups of patients and particularly in the critical group that eventually succumbed to the disease and died.  

IgG can take time to form after an infection and values were recorded in the three groups on the 14th day after getting symptoms

The results are recorded in the following table.

IgG (AU/ml)	Mild	Severe	Critical
	180 241 195 199 207 201 222 189 201	150 175 180 201 168 175 192 195	175 148 153 135 166
Means	203.8889	179.5	155.4
St-Dev	18.1207	16.4838	15.5981

Ho: Average levels of IgG antibody are the same in the groups.

H1: Average levels of IgG antibody are different in the groups.

(a) Complete the ANOVA table. Use α=0.05 significance level.

(Round the SS and Variance values to 2 decimal places and F-test-statistic to 1 decimal place. Leave no cells blank)

Source of Variation	Sums of Squares (SS)  (2dp)	DF	Variance (2dp)	F (1 dp)
Between Groups	7811.9111	Answer	Answer	Answer
Within Groups	Answer	Answer	Answer
Total	Answer	21

(b)	Look up the critical value from tables to answer the question below.
	There are Answer significant differences/not significant differences in the levels of IgG antibody across the groups.

(1 Mark)

Answer 1

(a)

The ANOVA Table is completed as follows:

Source of Variation	Sum of Squares	DF	Variance	F
Between groups	7811.9111	2	7811.9111/2=3905.96	3905.96/289.58=13.5
Within groups	5502.0889	19	5502.0889/19=289.58
Total	13314

The entries in the Table are got as follows:

From the given data the following Table is calculated:

	Mild	Severe	Critical
n	9	8	5
Sum	1835	1436	777
Mean	203.889	179.5	155.4
	376763	259664	121719
Std. Dev.	18.121	16.484	15.598
SS	2626.889	1902	973.2

(b)

Calculated value of F = 3905.9556/289.5836=13.5

Degrees of Freedom fornumerator = 2

Degrees of Freedom for denominator = 19

= 0.05

From Table, critical value of F =3.52

Since Calculated value of F =13.5 is greater than critical value of F = 3.52, the difference is significant. Reject null hypothesis.

Conclusion:

There are significant difference in the levels of IgG antibody across the groups.