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 16 grams of sugars: Sugars 95 \(\mathrm{P}\) \(\begin{array}{rrr}\text { rs Fit } & \text { SE Fit } \\\ 6 & 157.88 & 7.10 & \text { (143.3 }\end{array}\) \(95 \% \mathrm{Cl}\) 35,172.42) \(9 \%\) \(\begin{array}{lllll}16 & 15788 & 7.10 & (143.35,172.42) & (101.46\end{array}\) 214.31)

Short answer

The 95% confidence interval indicates that the true average calories for cereals with 16 grams of sugars is likely between 143.35 and 172.42. The 95% prediction interval allows us to predict that the calories in a single box of cereal with 16 grams of sugars will be between 101.46 and 214.31.

Step-by-step solution

Interpret the Confidence Interval

The 95% confidence interval given is (143.35, 172.42). This interval suggests that we can be 95% confident that the true average caloric content of all cereals with 16 grams of sugars lies within this range.

Interpret the Prediction Interval

The 95% prediction interval given is (101.46, 214.31). This means that we can predict with 95% certainty that the caloric content of an individual cereal box with 16 grams of sugars will fall within this range.

Understand The Difference Between Them

Note the difference between these two intervals. The confidence interval is about estimating the true population mean, while the prediction interval is about estimating individual data points.