Consider a random variable \(X\) that has a normal distribution with a mean of 5 and a variance of 9. Calculate \(P[X > 4]\).

`# your code here`

`# starter`

Consider the simple linear regression model

\[ Y = -3 + 2.5x + \epsilon \]

where

\[ \epsilon \sim N(0, \sigma^2 = 4). \]

What is the expected value of \(Y\) given that \(x = 5\)? That is, what is \(\text{E}[Y \mid X = 5]\)?

`# your code here`

Return to the simple linear regression model

\[ Y = -3 + 2.5x + \epsilon \]

where

\[ \epsilon \sim N(0, \sigma^2 = 4). \]

What is the standard deviation of \(Y\) when \(x\) is \(10\). That is, what is \(\text{SD}[Y \mid X = 10]\)?

`# your code here`

For this Exercise, use the built-in `trees`

dataset in `R`

. Fit a simple linear regression model with `Girth`

as the response and `Height`

as the predictor. What is the slope of the fitted regression line?

`# your code here`

For this Exercise, use the built-in `trees`

dataset in `R`

. Fit a simple linear regression model with `Girth`

as the response and `Height`

as the predictor. What is the value of \(R^2\) for this fitted SLR model?

`# your code here`

Consider the simple linear regression model

\[ Y = 10 + 5x + \epsilon \]

where

\[ \epsilon \sim N(0, \sigma^2 = 16). \]

Calculate the probability that \(Y\) is less than 6 given that \(x = 0\).

`# your code here`

Consider the simple linear regression model

\[ Y = 6 + 3x + \epsilon \]

where

\[ \epsilon \sim N(0, \sigma^2 = 9). \]

Calculate the probability that \(Y\) is greater than 1.5 given that \(x = -1\).

`# your code here`

Consider the simple linear regression model

\[ Y = 2 + -4x + \epsilon \]

where

\[ \epsilon \sim N(0, \sigma^2 = 25). \]

Calculate the probability that \(Y\) is greater than 1.5 given that \(x = 3\).

`# your code here`

For Exercises 9 - 15, use the `faithful`

dataset, which is built into `R`

.

Suppose we would like to predict the duration of an eruption of the Old Faithful geyser in Yellowstone National Park based on the waiting time before an eruption. Fit a simple linear model in `R`

that accomplishes this task.

What is the estimate of the intercept parameter?

`# your code here`

What is the estimate of the slope parameter?

`# your code here`

Use the fitted model to estimate the mean duration of eruptions when the waiting time is **78** minutes.

`# your code here`

Use the fitted model to estimate the mean duration of eruptions when the waiting time is **122** minutes.

`# your code here`

Consider making predictions of eruption duration for waiting times of 80 and 120 minutes, which is more reliable?

- 80
- 120
- Both are equally reliable

`# your code here`

Calculate the RSS for the fitted model.

`# your code here`

What proportion of the variation in eruption duration is explained by the linear relationship with waiting time?

`# your code here`

For this Exercise, use the built-in `trees`

dataset in `R`

.

Fit a simple linear regression model with `Girth`

as the response and `Height`

as the predictor. Use this fitted model to give an estimate for the mean `Girth`

of trees that are 81 feet tall.

Suppose both Least Squares and Maximum Likelihood are used to fit a simple linear regression model to the same data. The estimates for the slope and the intercept will be:

- The same
- Different
- Possibly the same or different depending on the data

Consider the fitted regression model:

\[ \hat{y} = -1.5 + 2.3x \]

Indicate all of the following that **must** be true:

- The difference between the \(y\) values of observations at \(x = 10\) and \(x = 9\) is \(2.3\).
- A good estimate for the mean of \(Y\) when \(x = 0\) is -1.5.
- There are observations in the dataset used to fit this regression with negative \(y\) values.

Indicate all of the following that are true:

- The SLR model assumes that errors are independent.
- The SLR model allows for a larger variances for larger values of the predictor variable.
- The SLR model assumes that the response variable follows a normal distribution.
- The SLR model assumes that the relationship between the response and the predictor is linear.

Suppose you fit a simple linear regression model and obtain \(\hat{\beta}_1 = 0\). Does this mean that there is **no relationship** between the response and the predictor?

- Yes
- No
- Depends on the intercept