This assignment has a total of 27 pts possible. Your score out of 24 will noted and scaled to 5 points (maximum of 5).
Part A (Study Design) What do we need to be true of a sample in order to generalize results to a population?
Part B (Study Design) Why does randomization of treatments to experimental units let us make causal conclusions?
Part C What form of relationship does Pearson’s correlation measure?
Part D If a scatterplot has a value of r = .9, does that mean there must be a linear relationship between the variables? Explain.
Part E What does the phrase ‘correlation \(\neq\) causation’ mean in your own words?
Part F What form of relationship needs to exist between quantitative variables to perform linear regression?
Part G Explain what extrapolation is and why we should probably try to avoid it.
IMS - Section 7.5, Question 7
The dataset below includes information on 31 black cherry trees felled in the Allegheny National Forest, Pennsylvania. For each tree, it includes three variables, one for each diameter (in), height (ft), and volume (cubic ft).
## Cherry tree data
cherry <- read.csv("https://collinn.github.io/data/cherry.csv")Part A: Create two scatterplots of the data comparing diameter with volume and height with volume, in each case letting volume be the response variable. Based on these plots, which variable do you think would be a better predictor of volume?
Part B: Create two linear models, ones for each of
the plots created in Part A (that is, with volume as a response variable
in both models). Based on the summary() output, which of
these models has a higher \(R^2\)
value? Is this consistent with what you decided in Part A?
Part C: Using the model with the highest \(R^2\) in Part B, write the linear equation for predicting a tree’s volume. Interpret both the slope and the intercept. Is the intercept meaningful in this case?
Part D: In 1 or two sentences, generally describe what the model in part C is doing.
Part E: Based on your \(R^2\) value for model in Part C, do you think a model using both variables will perform much better than the regression model just using one predictor?
Part F: Make a regression model using both predictors. Interpret the coefficients. Do you think the increase in \(R^2\) warrants the increase in model complexity adding another variable?
The problem includes a dataset with 144 cats, included with each observation is the sex of the cat, as well as body weight (kg) and heart weight (g).
## Read in cat data
cats <- read.csv("https://collinn.github.io/data/cats.csv")Part A: Use lm() to create a linear
model in R predicting the weight of a cat’s heart using body weight as
an explanatory variable. Write the formula for the regression line in
context.
Part B: Interpret the slope in context.
Part C: Interpret the intercept in context. Is the intercept meaningful?
Part D: What is the predicted heart weight of a cat that has a body weight of 3kg?
Part E: What is the residual for a cat with a body weight of 3kg that actually has a heart weight of 12g. Is this an under- or over-prediction?
Part F: Create a linear model in R predicting the weight of a cat’s heart using the cat’s sex as an explanatory variable. Write the formula for the regression equation. Interpret the coefficients.
Part G: Create a third linear model, this time including the cat’s sex in addition to body weight to predict heart weight. How do we interpret the intercept in this model?
Part F: Using the model from part C, what heart weight would you predict for:
a male cat with a body weight of 3.2kg
a female cat with a body weight of 2.4kg