Question

FIBER IN CEREALS AS A PREDICTOR OF CALORIES In Example 9.10 on page \(592,\) we look at a model to predict the number of calories in a cup of breakfast cereal using the number of grams of sugars. In Exercises 9.64 and 9.65 , we give computer output with two regression intervals and information about a specific amount of sugar. 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 cereals with 10 grams of sugars: \(\begin{array}{rrrrr}\text { Sugars } & \text { Fit } & \text { SE Fit } & & 95 \% \text { Cl } & 95 \% \text { PI } \\ 10 & 132.02 & 4.87 & (122.04,142.01) & (76.60,187.45)\end{array}\)

Short answer

The \(95 \%\) confidence interval for the mean response, between 122.04 and 142.01 calories, indicates the range where the true average calorie count for cereals with 10 grams of sugar is likely to fall, with \(95 \%\) confidence. The \(95 \%\) prediction interval for an individual response, between 76.60 and 187.45 calories, indicates the range where the calorie count of a single cup of cereal with 10 grams of sugar is likely to fall, with \(95 \%\) confidence.

Step-by-step solution

Interpret Confidence Interval

The \(95\%\) confidence interval for the mean response refers to the interval within which we are \(95\%\) certain that the true mean response (mean number of calories for cereals with 10g of sugars) lies. Here it is interpreted as: We can be \(95\%\) confident that the average calorie content of cereals with 10g of sugars lies between 122.04 and 142.01 calories.

Interpret Prediction Interval

The \(95\%\) prediction interval for an individual response refers to the interval within which we are \(95\%\) certain that the response (number of calories in a single cup of cereal with 10g of sugars) will fall. Here it is interpreted as: We can predict with \(95\%\) confidence that the number of calories in a single cup of cereal with 10g of sugars will lie between 76.60 and 187.45 calories.

Recognize the differences

It's important to note that the prediction interval is wider than the confidence interval, indicating more uncertainty around predicting a single response versus the mean response. While the confidence interval predicts where the mean response will likely fall, the prediction interval accounts for individual data point variability.

Key concepts

Confidence Interval

In statistics, confidence intervals play a pivotal role in understanding the precision of an estimated population parameter. For instance, when we talk about the 95% confidence interval for the mean response in the context of our cereal example, we refer to a range within which we can say with 95% certainty that the true average number of calories for cereals with 10 grams of sugar will fall. The confidence interval conveys a level of assurance in our estimate based on the data obtained and the statistical method used.

To break it down, the lower and upper limits of the interval—122.04 and 142.01 calories, respectively—form a band around the estimated mean (132.02 calories). This means that if we were to repeat our study multiple times, collecting different samples each time, the true mean would lie within this interval 95% of the time. This interval, however, does not predict individual responses but focuses solely on the central tendency of the population being studied.

Understanding confidence intervals allows students to appreciate the uncertainty inherent in any estimate, reinforcing the idea that statistics often deals with probabilities rather than absolutes. Proper interpretation of such intervals fosters informed decision-making in data analysis.

Prediction Interval

When we step from estimating a population parameter to predicting individual observations, we turn to prediction intervals. In our exercise, the 95% prediction interval for the response provides us with a range where individual cereal cups' calorie count is expected to fall, given they contain 10 grams of sugar. While both intervals incorporate uncertainty, the prediction interval is broader (76.60 to 187.45 calories) because it accounts for the variability of each individual observation.

Why is it wider? Imagine we're looking not just at averages but at the dispersion of all possible cereal cups out there. Each cup might differ slightly due to various uncontrollable factors. As a result, when predicting a single observation, we must account for more uncertainty than when predicting a mean.

Students should realize that prediction intervals will invariably be larger than confidence intervals because they take into account the individual variation as well as the uncertainty of estimating the mean. Recognizing this difference is crucial when making predictions about future data points and can prevent the common mistake of underestimating the potential variance in predictions.

Mean Response Prediction

Mean response prediction involves forecasting the average outcome—like calorie content for cereal based on sugar content—based on a regression model. It is central to understanding the relationship between a predictor variable and an outcome variable. In the context of our example, the regression model allows us to predict that the average number of calories in cereals with 10 grams of sugar would be about 132.02 calories. This figure, known as the 'Fit' in our table, is an estimate derived from the sample data using regression analysis.

The concept of mean response prediction is instrumental in numerous fields where determining the expected outcome is essential. By examining relationships between variables, we can make informed predictions that serve as the baseline for policies, scientific studies, and business decisions. Through this predictive power, we gain insights that guide our expectations and strategies in the face of future data.