## #1

The test statistic for $$H_0: \beta_1 = \beta_2 = 0$$ is the $$F$$ statistic. It's what we'll use for when we're testing multiple parameters at once.

Several people had $$\beta_1 = 0$$ or $$\beta_2 = 0$$. This is wrong; it should be and not or. This is actually an important difference since the distribution for the $$F$$ statistic is testing for both slopes simultaneously being zero.

## #2

This one is similar to #1, except we're testing a single parameter. That means we want the $$t$$ statistic. We want the $$t$$ from the full model, since we're interested in if winterizing is a significant predictor after accounting for thermostat setting.

## #3

Standard interpretation for a multiple regression slope. Make sure to explain that this is the slope for Winter, when holding Therm constant.

## #4

Similar to #3, but from the simple regression model. We know we want the simple regression since we're asked for the total effect (not controlling for anything else).

Some people were mixing up the response and the predictor variables in the interpretation. Remember, the slope is $$\frac{\Delta y}{\Delta x}$$. We change $$x$$ (the predictor) and see how $$y$$ responds. To keep the interpretation simple, we change $$x$$ by one unit. Then the change in $$y$$ is the estimated slope $$\hat{\beta}$$.

## #5

The direct effect is the slope from the full regression model. It's the direct effect since we've controlled for all the other predictors, there's nothing else included in the slope.

# Section B42

## #1

The test statistic for $$\beta_1 = \beta_2 = \beta_3 = 0$$ is the $$F$$ statistic. It's what we'll use for when we're testing multiple parameters at once.

Several people had $$\beta_1 = 0$$ or $$\beta_2 = 0$$ or $$\beta_3 = 0$$. This is wrong; it should be and not or. This is actually an important difference since the distribution for the $$F$$ statistic is testing for both slopes simultaneously being zero.

## #2

Be careful to not mix up interpreting slopes vs. interpreting tests. #2 asked for you to interpret the result of a $$t$$ test: is $$X_2$$ is significant predictor of $$y$$, after controlling for the other predictors?

## #3

Make sure you know what all you're controlling for when interpreting slopes. For this one the best conservative model only had HSM as a predictor. That means our interpretation of $$\hat{B_1}$$ doesn't include controlling for the other predictors.

For the prediction in part c, a lot of people made the same mistake as last week: they reported the point estimate $$\hat{y}$$, instead of an interval. The question asked us to estimate with 90% certainty, so we know the answer should be an interval.

## #4

The population regression equation is

$$y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_3 + \epsilon$$

where the $$\beta_i$$'s are unknown. We want to slice out the subsection of the population with the grades given in the question. Substitute those in for the $$X_i$$'s. This is still a population regression equation (but for a subset of the population), so the $$\beta_i$$'s are still unknown.