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