The following model was fitted to a sample of 25 students using data obtained at the end of their freshman year in college. The aim was to explain students' weight gains:
$$
y=\beta_0+\beta_1 x_1+\beta_2 x_2+\beta_3 x_3+\varepsilon
$$
where
$y=$ weight gained, in pounds, during freshman year
$x_1=$ average number of meals eaten per week
$x_2=$ average number of hours of exercise per week
$x_3=$ average number of beers consumed per week
The least squares estimates of the regression parameters were as follows:
$$
b_0=7.35 \quad b_1=0.653 \quad b_2=-1.345 \quad b_3=0.613
$$
The estimated standard errors were as follows:
$$
s_{b_1}=0.189 \quad s_{b_2}=0.565 \quad s_{b_3}=0.243
$$
a. Test, against the appropriate one-sided alternative, the null hypothesis that, all else being equal, hours of exercise do not linearly influence weight gain.
b. Test, against the appropriate one-sided alternative, the null hypothesis that, all else being equal, beer consumption does not linearly influence weight gain.
c. Find $90 \%, 95 \%$, and $99 \%$ confidence intervals for $\beta_1$.