HW 2 Notes

a.

$${\hat{\beta }}_0,{\hat{\beta }}_1$$ $$s.e({\hat{\beta }}_0)s.e({\hat{\beta }}_1)$$

b.

$${\hat{\beta }}_0=1.17$$ $${\hat{\beta }}_1=1.72$$ $${\hat{y}}_i=1.17+1.72x_i$$

When x =0 the average national income would be 1.17 million

c. classical assumption means all other assumptions hold. (3 other assumptions are true, find these later)

$$H_0={\beta }_1=0$$

vs

$$H_1={\beta }_1{\ne }_0$$

Under $H_0$

$$t=\frac{{\hat{\beta }}_0-{\hat{\beta }}_1}{s.e({\hat{\beta }}_1)}=\frac{{\hat{\beta }}_1}{s.e({\hat{\beta }}_1)}\approx t_{n-2}=t_8$$ $$t_{actual}=\frac{{\hat{\beta }}_1}{s.e({\hat{\beta }}_1)}=\frac{1.72}{?}=?$$

now $\alpha$ = 5%

$$c.v=t_8,0.025=2.306$$

if

$$|t_{act}|>c.v=2.306$$

We reject $H_{0}$ at 5%

S.L based on the statistical evidence

Can also use p value approach

$$P-val=$$

e

$$R^2=1-\frac{{\sum }_{i=1}^n{\hat{\Sigma }}_i^2}{{\sum }_{i=1}^n(Y_i-\bar{Y}{)}^2}$$

f write down general formula

$$100(1-\alpha )\%C.I$$

for ${\beta }_1$