## #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

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.