Consider a random variable \(X\) that has a \(t\) distribution with \(7\) degrees of freedom. Calculate \(P[X > 1.3]\).

`# your code here`

Consider a random variable \(Y\) that has a \(t\) distribution with \(9\) degrees of freedom. Find \(c\) such that \(P[X > c] = 0.025\).

`# 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 p-value for testing \(H_0: \beta_1 = 0\) vs \(H_1: \beta_1 \neq 0\)?

`# your code here`

Continue using the SLR model you fit in Exercise 3. What is the length of a 90% confidence interval for \(\beta_1\)?

`# your code here`

Continue using the SLR model you fit in Exercise 3. Calculate a 95% confidence interval for the mean tree girth of a tree that is 79 feet tall. Report the upper bound of this interval.

`# your code here`

Consider a random variable \(X\) that has a \(t\) distribution with \(5\) degrees of freedom. Calculate \(P[|X| > 2.1]\).

`# your code here`

Calculate the critical value used for a 90% confidence interval about the slope parameter of a simple linear regression model that is fit to 10 observations. (Your answer should be a positive value.)

`# your code here`

Consider the true simple linear regression model

\[ Y_i = 5 + 4 x_i + \epsilon_i \qquad \epsilon_i \sim N(0, \sigma^2 = 4) \qquad i = 1, 2, \ldots 20 \]

Given \(S_{xx} = 1.5\), calculate the probability of observing data according to this model, fitting the SLR model, and obtaining an estimate of the slope parameter greater than 4.2. In other words, calculate

\[ P[\hat{\beta}_1 > 4.2] \]

`# your code here`

For Exercises 9 - 13, 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 value of \(\text{SE}[\hat{\beta}_1]\)?

`# your code here`

What is the value of the test statistic for testing \(H_0: \beta_0 = 0\) vs \(H_1: \beta_0 \neq 0\)?

`# your code here`

What is the value of the test statistic for testing \(H_0: \beta_1 = 0\) vs \(H_1: \beta_1 \neq 0\)?

`# your code here`

Test \(H_0: \beta_1 = 0\) vs \(H_1: \beta_1 \neq 0\) with \(\alpha = 0.01\). What decision do you make?

- Fail to reject \(H_0\)
- Reject \(H_0\)
- Reject \(H_1\)
- Not enough information

`# your code here`

Calculate a 90% confidence interval for \(\beta_0\). Report the upper bound of this interval.

`# your code here`

For this Exercise, use the `Orange`

dataset, which is built into `R`

.

Use a simple linear regression model to create a 90% confidence interval for the change in mean circumference of orange trees in millimeters when age is increased by 1 day. Report the lower bound of this interval.

`# your code here`

For this Exercise, use the `Orange`

dataset, which is built into `R`

.

Use a simple linear regression model to create a 90% confidence interval for the mean circumference of orange trees in millimeters when the age is 250 days. Report the lower bound of this interval.

`# your code here`

For this Exercise, use the `cats`

dataset from the `MASS`

package.

Use a simple linear regression model to create a 99% prediction interval for a catâ€™s heart weight in grams if their body weight is 2.5 kilograms. Report the upper bound of this interval.

```
library(MASS)
# your code here
```

Consider a 90% confidence interval for the mean response and a 90% prediction interval, both at the same \(x\) value. Which interval is narrower?

- Confidence interval
- Prediction interval
- No enough information, it depends on the value of \(x\)

Suppose you obtain a 99% confidence interval for \(\beta_1\) that is \((-0.4, 5.2)\). Now test \(H_0: \beta_1 = 0\) vs \(H_1: \beta_1 \neq 0\) with \(\alpha = 0.01\). What decision do you make?

- Fail to reject \(H_0\)
- Reject \(H_0\)
- Reject \(H_1\)
- Not enough information

Suppose you test \(H_0: \beta_1 = 0\) vs \(H_1: \beta_1 \neq 0\) with \(\alpha = 0.01\) and fail to reject \(H_0\). Indicate all of the following that must always be true:

- There is no relationship between the response and the predictor.
- The probability of observing the estimated value of \(\beta_1\) (or something more extreme) is greater than \(0.01\) if we assume that \(\beta_1 = 0\).
- The value of \(\hat{\beta}_1\) is very small. For example, it could not be 1.2.
- The probability that \(\beta_1 = 0\) is very high.
- We would also fail to reject at \(\alpha = 0.05\).

Consider a 95% confidence interval for the mean response calculated at \(x = 6\). If instead we calculate the interval at \(x = 7\), mark each value that would change:

- Point Estimate
- Critical Value
- Standard Error