Question

Observations are taken on net revenue from sales of a certain plasma TV at 30 retail outlets. A linear regression model was formed using the following variables: Y = net revenue (thousands of dollars); X₁ = shipping cost (dollars per unit); X₂ = expenditures on print advertising (thousands of dollars); and X₃ = expenditures on electronic media ads (thousands of dollars). Partial regression output appears below.

variables	coefficient	std. error	t-value	p-value
Intercept ShipCost PrintAds WebAds	4.31 -0.08 2.26 2.49	70.82 4.67 1.05 -	0.06 -0.02 2.15 2.96	0.95 0.98 0.03 0.004

Picture all of the components that go into the construction of a 90% confidence interval for the coefficient on PrintAds. For this problem, calculate the lower 90% confidence interval limit on the true β for PrintAds. Carry to four decimals.

Answer 1

Note that

$t=\frac{b-\beta}{s.e(b)}\sim t_{n-4},here\ n=30\\ b=estimate\ of\ \beta\ ,s.e(b)=standard\ error\ of\ b$.

Here we need to find 90% CI of $\beta$ .

So the 90% CI is as follows

$P\left ( c_1<\frac{b-\beta}{s.e(b)}<c_2 \right )=0.9(c_1<c_2)\\ \Rightarrow (c_1<t_{26}<c_2)=0.9 \Rightarrow c_1=t_{26,0.05},c_2=-t_{26,0.05},t_{26,0.05}=-1.705618\\ so\ 90\% \ CI\ is\\ \left ( b- t_{26,0.05}{s.e(b)},b+t_{26,0.05}{s.e(b)}\right )\\ \equiv(0.4691,4.0509)$