• 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

As you’ve seen previously for means, proportions, and slope, you can calculate an interval to account for the variability in the predicted values. 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 types of ways we can use the equation of the line of best fit:

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 whether the goal is to estimate the mean response for a value of the explanatory variable or to predict the value of the response variable for an individual observation.

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. This interval gives us a range of plausible values the mean value of the response variable takes when [latex]x=x_0[/latex].

We can interpret the [latex]C\%[/latex] interval as follows: We are [latex]C\%[/latex] confident that the mean response when the explanatory variable equals [latex]x_0[/latex] is between (lower bound) and (upper bound).

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]x_0[/latex] is the confidence level. This interval gives us a range of plausible values of the response when an individual observation has a value of the explanatory variable equal to [latex]x_0[/latex].

We can interpret the interval as follows: We are [latex]C\%[/latex] confident that the value of the response variable for an individual with a value of the explanatory variable equal to [latex]x_0[/latex] is between (lower bound) and (upper bound).

Prediction Interval vs. Confidence Interval^[1]

While you can use both prediction intervals and confidence intervals to quantify uncertainty in statistical analysis, it is important to understand how they differ from each other so you can choose the best one to use for each situation. Here are some key differences between the prediction interval and the confidence interval:

• A prediction interval includes a wider range of values than a confidence interval.
• A prediction interval is less certain than a confidence interval.
• A prediction interval predicts an individual number, whereas a confidence interval predicts the mean value.
• A prediction interval focuses on future events, whereas a confidence interval focuses on past or current events.

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