16C Preview

Preparing for the next classIn the next class, you will need to use technology to calculate predictions from a regression line, confidence intervals for the mean response, and prediction intervals for individuals. You will also need to identify the appropriate type of interval based on a given situation and compare confidence and prediction intervals. In the next in-class activity, you will analyze predictions from the linear regression equation. In practice, it is not sufficient to provide a single point estimate; therefore, you will also learn how to calculate intervals from the predicted values as well. This concept is similar to the intervals calculated for the mean in previous in-class activities.This assignment will help you review the skills you’ll need for the in-class activity and introduce intervals for two types of predictions in linear regression.How much do we expect a penguin to weigh based on the length of its flipper? To answer this question, we will look at the “Penguins”dataset.^[1] It contains characteristics fora sample of 342 penguins observed near Palmer Station, Antarctica. We will focus on the following two variables:

•flipper_length_mm: Flipperlength in millimeters

•body_mass_g: Body mass(weight)in grams

Question 1

1) Before taking flipper length into account, let’s start by looking at the mean body mass for penguins. The 90% confidence interval for the mean body mass is (4130, 4273). Select the correct interpretation of the interval.

a) We are 90% confident that the mean body mass of penguins is between 4,130 and 4,273 grams.

b) There is a 90% probability that the mean body mass of penguins is between 4,130 and 4,273 grams.

c) There is a 90% probability that the body mass of a penguin is between 4,130 and 4,273 grams.

d) Based on this interval,90% of penguins have a body mass between 4,130 and 4,273 grams.

Question 2

2) Now let’s take flipper length into account so we can get an estimate of the body mass of a penguin based on its flipper length. Go to the DCMPLinear Regression tool at https://dcmathpathways.shinyapps.io/LinearRegression/ to calculate the equation of the line of best fit.

•Access spreadsheet DCMP_STAT_16C_Penguins.

•Under “Enter Data,”select “Enter Own.”

•Select the appropriate explanatory variable (𝑋) and responsevariable (𝑌).

•Enter the data.

Part A: Write the equation of the lineof best fitusing contextualized variablenames.

Part B: Suppose you know the flipper lengthsbut don’t know the body massesof two penguins. Penguin A has a flipper length of 202millimeters (mm)and Penguin B has a flipper length of 172 mm. Complete the followingstatement.We expect the body mass of Penguin B to be about _____ grams ______(less/greater) than the body mass of Penguin A.

Part C: Calculate the predicted body mass for each penguin. Round your answers to one decimal place.

Though the regression equation is used to calculate the expected body mass given the flipper length, we know that multiple penguins can have the same flipper length and different body masses. (This occurs quite frequently in our dataset!) Therefore, if we are trying to predict the weight of an individual penguin, it makes sense to calculate an interval that takes the variability in the actual penguin weights into account. In addition, thinking about what we have learned about sample variability in previous activities, we know that if we randomly select another sample with 324 penguins, the equation of the line of best line will be different—so the predicted body mass for a given flipper length (the point estimate) will change. Before calculating the interval for predicted values, however, we need to first consider the type of prediction we’re most interested in obtaining. There are two ways we can use the linear regression equation:

1. To estimate the mean value of the response when the explanatory variable is equal to a particular value, 𝑥0

2. To predict the value of the response for an individual observation when the explanatory variable is equal to 𝑥0

The type of interval calculated will depend on how we want to use the linear regression equation.

When the objective is to estimate the mean value of the response variable for a particular value of the explanatory variable, 𝑥0, we will calculate a confidence interval for the mean response, where 𝐶 is the confidence level associated with the interval. This interval gives us a range of plausible values of the mean response for the subset of the population with a value of the explanatory variable equal to𝑥0.

Question 3

3) We will use technology to calculate the following interval. During the in-class activity, you will learn more about these intervals and how to interpret them.

Part A: Calculate a 92% confidence interval for the mean body mass for penguins with a flipper length of 202 mm, the same flipper length as Penguin A. In theDCMP Linear Regression tool, under theFit Linear Regression Model tab ,you can obtain the interval by checking “Confidence/Prediction Interval” in the toolbar on the left-hand side and inputting the value of the explanatory variable under“x-value.” Select the appropriate level of confidence, 𝐶, by moving the slider. When the objective is to predict the value of the response variable for an individual observation with the explanatory variable equal to 𝑥0, we will calculate a 𝑪% prediction interval for an individual response, where 𝐶 is the confidence level associated with the interval. This interval gives us a range of plausible values of the response for an individual observation that has a value of the explanatory variable equal to𝑥0.

Part B: Use the linear regression tool(i.e., data analysis tool)to calculate a 92% prediction interval for the body mass of Penguin A, who has a flipper length of 202 mm.

Hint: Check the box “For Individual Response.”

Question 4

4) For each scenario, determine whether a confidence or prediction interval is most appropriate. Then use the linear regression tool (i.e., data analysis tool) to calculate the appropriate interval. Penguin B has a flipper length of 172 mm.

Part A: Calculate the predicted body mass and corresponding interval for Penguin B from Question 2.

Part B: Calculate the predicted mean body mass and corresponding interval for penguins with flippers that are 172 mm long.

Question 5

5) Let’s use the intervals from Questions 3 and 4 to describe the relationship between the confidence interval for the mean response and the prediction interval for an individual observation.

Part A: Given a value of the explanatory variable, how do the centers of the confidence and prediction intervals compare?

a) The center of the confidence interval is less than the center of the prediction interval.

b) The center of the confidence interval is equal to the center of the prediction interval.

c) The center of the confidence interval is greater than the center of the prediction interval.

Part B: Given a value of the explanatory variable, how do the widths of the confidence and prediction intervals compare?

a) The width of the confidence interval is less than the width of the prediction interval.

b) The width of the confidence interval is equal to the width of the prediction interval.

c) The width of the confidence interval is greater than the width of the prediction interval.

Question 6

6) Based on the work you’ve done in this preview assignment, why do you think it’s important to include the confidence/prediction intervals when reporting predictions from the regression line, rather than relying only on the single predicted value? Provide one to two reasons.