• Find and interpret the confidence interval for the mean response
• Find and interpret the prediction interval for an individual response
• Identify whether a confidence interval or a prediction interval is more appropriate in context of the problem

Intervals for the Mean Response

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 data set!) 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 [latex]324[/latex] 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, [latex]x_0[/latex]
2. To predict the value of the response for an individual observation when the explanatory variable is equal to [latex]x_0[/latex]

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

confidence interval for the mean response

When the objective is to estimate the mean value of the response variable for a particular value of the explanatory variable, [latex]x_0[/latex], we will calculate a confidence interval for the mean response, where [latex]x_0[/latex] 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 to [latex]x_0[/latex].

prediction interval for an individual response

When the objective is to predict the value of the response variable for an individual observation with the explanatory variable equal to [latex]x_0[/latex], we will calculate a [latex]C[/latex]% prediction interval for an individual response, where [latex]C[/latex] 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 to [latex]x_0[/latex].

Step 1: Access spreadsheet Penguins.
Step 2: Under “Enter Data,” select “Enter Own.”
Step 3: Copy and paste the appropriate explanatory variable ([latex]x[/latex]) and response variable ([latex]y[/latex]).
Step 4: Check the “Confidence/Prediction Interval” and input the value of the explanatory variable under “x-value.” Select the appropriate level of confidence, [latex]C[/latex], by moving the slider.

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