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# starterConsider 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 hereReturn 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 hereFor 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 hereFor 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 hereConsider 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 hereConsider 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 hereConsider 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 hereFor 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 hereWhat is the estimate of the slope parameter?
# your code hereUse the fitted model to estimate the mean duration of eruptions when the waiting time is 78 minutes.
# your code hereUse the fitted model to estimate the mean duration of eruptions when the waiting time is 122 minutes.
# your code hereConsider making predictions of eruption duration for waiting times of 80 and 120 minutes, which is more reliable?
# your code hereCalculate the RSS for the fitted model.
# your code hereWhat proportion of the variation in eruption duration is explained by the linear relationship with waiting time?
# your code hereFor 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:
Consider the fitted regression model:
\[ \hat{y} = -1.5 + 2.3x \]
Indicate all of the following that must be true:
Indicate all of the following that are true:
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?