## Exercise 1

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

# your code here

## Exercise 2

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

## Exercise 3

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

## Exercise 4

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

## Exercise 5

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

## Exercise 6

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

# your code here

## Exercise 7

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

## Exercise 8

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

## Exercise 9

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

## Exercise 10

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

## Exercise 11

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

## Exercise 12

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

## Exercise 13

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

# your code here

## Exercise 14

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

## Exercise 15

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

## Exercise 16

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 cat’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

## Exercise 17

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 $$x$$

## Exercise 18

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

## Exercise 19

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 $$\alpha = 0.05$$.

## Exercise 20

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