Question

An experiment was conducted to study the effect of pesticides on plant species that are exposed to them. Different dosages of pesticides were administered to 4 groups of 4 mice. The 16 mice were females of similar age and condition. One group received no pesticides. The response Y was measure of brain activity. It was postulated that brain activity (Y in moles/liters per minute) would decrease with an increase in dosage (X in mg/kg body weight)

Dose (X)

0.0, 0.0, 0.0, 0.0, 2.5, 2.5, 2.5, 2.5

Activity (Y)

10.6, 10.4, 10.8, 11.0, 11.0, 11.3, 10.3, 9.9

Dose (X)

5.0, 5.0, 5.0, 5.0, 7.5, 7.5, 7.5, 7.5

Activity (Y)

9.5, 9.2, 9.7, 8.6, 8.2, 8.0, 8.4, 7.8

a) test if there is a significant linear relationship between the dose and the brain activity by constructing an analysis of variance table for the above data.

b) compute r and interpret the result.

Answer 1

a)

Regression ANOVA:

$H_0:Y=b_0$ , there is no a significant linear relationship between the dose and the brain activity 

$H_a:Y=b_0+b_1X$ , there is a significant linear relationship between the dose and the brain activity

Regression line equation:

$Ŷ = b_0 +b_1X$

$b_1 = \frac{ Σ(x_i-x̄)(y_i-ȳ)}{ Σ(x_i-x̄)^2}=-0.367$

$b_00 = ȳ - b_1x̄=11.045$

$Ŷ = 11.045 - 0.367X$

for Regression:

df = p(number of variables) - number of independent variables = 1

Sum of Square:

$SS_{regression}=\sum (\tilde{y}_i-\overline{y})^2=16.8361$

where $\tilde{y}_i$ are predicted values

Mean Square:

$MS_{regression}=SS_{regression}/(p-1)=16.8361$

for Residual:

$df=n-1-1=14$

n is total number of observations

Sum of Square:

$SS_{residual}=\sum (y_i-\tilde{y}_i)^2=3.9382$

Mean Square:

$MS_{residual}=SS_{residual}/(n-1-1)=0.2813$

F Statistic:

$F=MS_{regression}/MS_{residual}=59.85$

critical value for $\alpha=0.05$ :

$F_{(1,14)}=4.6001$

Since $F>F_{(1,14)}$ we reject null hypothesis. There is a significant linear relationship between the dose and the brain activity.

b.

correlation coefficient:

$r=\frac{Σ(x_i - x̄)(y_i - ȳ)}{ \sqrt{(Σ(x_i - x̄)^2Σ(y_i - ȳ)^2 )}} =-0.9002$

there is a very strong inverse relationship between the dose and the brain activity