An aircraft company wanted to predict the number of worker-hours necessary to finish the design of a new plane. Relevant explanatory variables were thought to be the plane's top speed, its weight, and the number of parts it had in common with other models built by the company. A sample of 27 of the company's planes was taken, and the following model was estimated:
$$
y=\beta_0+\beta_1 x_1+\beta_2 x_2+\beta_3 x_3+\varepsilon
$$
where
$y=$ design effort, in millions of worker-hours
$x_1=$ plane's top speed, in miles per hour
$x_2=$ plane's weight, in tons
$x_3=$ percentage of parts in common with other models
The estimated regression coefficients were as follows:
$$
b_1=0.661 \quad b_2=0.065 \quad b_3=-0.018
$$
The estimated standard errors were as follows:
$$
s_{b_1}=0.099 \quad \mathrm{~s}_{b_2}=0.032 \quad \mathrm{~s}_{b_3}=0.0023
$$
a. Find $90 \%$ and $95 \%$ confidence intervals for $\beta_1$.
b. Find $95 \%$ and $99 \%$ confidence intervals for $\beta_2$.
c. Test against a two-sided alternative the null hypothesis that, all else being equal, the plane's weight has no linear influence on its design effort.
d. The error sum of squares for this regression was 0.332 . Using the same data, a simple linear regression of design effort on the percentage of common parts was fitted, yielding an error sum of squares of 3.311. Test, at the $1 \%$ level, the null hypothesis that, taken together, the variable's top speed and weight contribute nothing in a linear sense to explaining the changes in the variable, design effort, given that the variable percentage of common parts is also used as an explanatory variable.