Question

In Exercises 9.62 and 9.63 , we give computer output with two regression intervals and information about the percent of calories eaten during the day. Interpret each of the intervals in the context of this data situation. (a) The \(95 \%\) confidence interval for the mean response (b) The \(95 \%\) prediction interval for the response The intervals given are for mice that eat \(50 \%\) of their calories during the day: \(\begin{array}{rrrrr}\text { DayPct } & \text { Fit } & \text { SE Fit } & 95 \% \mathrm{Cl} & 95 \% \mathrm{PI} \\ 50.0 & 7.476 & 0.457 & (6.535,8.417) & (2.786,12.166)\end{array}\)

Short answer

The 95% confidence interval for the mean response (6.535, 8.417) indicates that we are 95% confident that the true mean response for all mice eating 50% of their calories during the day lies within this range. The 95% prediction interval for the response (2.786,12.166) indicates that we are 95% confident that the true response for a single mouse eating 50% of its calories during the day will fall within this range.

Step-by-step solution

Understanding the Data Set

First, understand where this data comes from. This is regression output concerning mice eating habits. The variable 'DayPct' refers to the percentage of calories the mice consume during the day. The 'Fit' value is a predicted response from the regression model, while 'SE Fit' refers to the standard error of the fit. The 95% Cl and 95% PI are the confidence and prediction intervals respectively, generated by the regression model, both for when 'DayPct' is 50.0.

Interpreting Confidence Interval

Now, interpret the confidence interval. The interval given here is (6.535,8.417). It means that, assuming our model is correct, we are 95% confident that the true mean response of all mice that eat 50% of their calories during the day lies within this interval.

Interpreting Prediction Interval

Finally, interpret the prediction interval, which is given here as (2.786,12.166). This means that, assuming our model is correct, we are 95% confident that the true response for an individual mouse that eats 50% of its calories during the day will fall within this interval.

Key concepts

Confidence Interval

When dealing with statistical analyses, a confidence interval (CI) is a range of values that's used to estimate the true value of a population parameter. In the context of regression analysis, a CI gives us an idea of where the true mean response is expected to fall based on the sample data. For instance, the 95% CI, such as the one illustrated in the exercise for mice eating habits, tells us that we can be 95% confident that the interval (6.535, 8.417) contains the true mean number of calories mice consume during the day when 'DayPct' is 50 percent.

Why is it 95% and not 100% confident? Because it accounts for variability and acknowledges that there is always a 5% chance that the interval does not contain the true mean. CIs are useful for assessing the reliability of an estimated mean; wider intervals suggest more variability and less certainty. The calculation of a CI incorporates the standard error, the sample size, and the desired level of confidence to give researchers a useful measure of precision in their estimates.

Interpreting the CI in our textbook exercise, it suggests that the researchers are confident that the average caloric intake during the day for all mice (not just the sample) that consume 50% of calories during daytime is between 6.535 and 8.417.

Prediction Interval

Prediction intervals (PI) share some resemblance with confidence intervals, but they serve a distinctly different purpose in regression analysis. While CIs provide a range for the mean value of the dependent variable across the population, PIs give a range where we can expect to observe future individual data points. Specifically, in our textbook exercise, the 95% prediction interval of (2.786, 12.166) for mice eating 50% of their calories during the day gives us a range in which we expect the actual caloric intake of any one mouse to fall.

The PI is wider than the CI because it has to account for not only the uncertainty in the estimate of the mean but also the individual random variation. PIs are considerably valuable when making predictions for individual observations, as it underlines the inherent unpredictability of such forecasts. They are especially important in settings where decisions are made on an individual basis, like personalized medicine or tailored nutritional recommendations for the mice based on their eating habits.

Regression Analysis

At the heart of this textbook exercise is regression analysis, a powerful statistical method used to examine the relationship between a dependent variable and one or more independent variables. The primary goal is to model the expected value of the dependent variable based on the values of the independent ones. In simple terms, it helps us understand how much the dependent variable changes when any one of the independent variables is varied, while the others are held fixed.

In the context of the given data, regression analysis helps us understand how the percent of calories mice eat during the day ('DayPct') relates to their overall caloric intake ('Fit'). Regression models generate coefficients that tell us the direction and magnitude of the relationship between variables. Moreover, these models facilitate the creation of both confidence and prediction intervals, which are essential for interpreting the data and making forecasts. By estimating the true relationship between variables, regression analysis becomes a cornerstone of predictive analytics.

Standard Error

An indispensable concept in regression analysis is the standard error (SE). This metric captures the average distance that the observed data points deviate from the regression line. SE helps us understand the accuracy of our predictions; a smaller SE indicates that the data points are closer to the predicted values and suggest a better fit of the model.

In our exercise, 'SE Fit' refers to the standard error of the fit for the regression model — in simple terms, it's an estimate of the standard deviation of the random component of the data. The standard error is critically involved in the calculation of confidence intervals, where a small standard error can lead to a narrower confidence interval, implying a more precise estimate of the population mean. Similarly, it affects the prediction interval width, with a smaller SE leading to a tighter range of predicted values for an individual observation. Understanding the standard error can give students insights into the reliability of the estimated coefficients and the overall quality of the regression model.