[DATA] The following data represent the height (inches) of boys between the ages of 2 and 10 years.

h. Explain why the predicted heights found in parts (a) and (f) are the same, yet the intervals are different.

Verified step by step guidance

1. Step 1: Understand that the predicted heights in parts (a) and (f) come from the same regression equation, which models the relationship between boy's age (x) and height (y). Since the prediction is made for the same age value, the predicted height (the point estimate) will be the same in both parts.
2. Step 2: Recall that the regression equation is generally of the form \(\hat{y} = b_0 + b_1 x\), where \(b_0\) is the intercept and \(b_1\) is the slope. The predicted height \(\hat{y}\) is calculated by substituting the given age \(x\) into this equation.
3. Step 3: Recognize that the difference in intervals arises because the intervals in parts (a) and (f) represent different types of confidence or prediction intervals. For example, one might be a confidence interval for the mean height at a given age, and the other might be a prediction interval for an individual boy's height at that age.
4. Step 4: Understand that a confidence interval for the mean height at a given age is narrower because it estimates the average height of all boys at that age, while a prediction interval is wider because it accounts for individual variation around that mean.
5. Step 5: Summarize that although the predicted height (point estimate) is the same in both parts, the intervals differ due to the different purposes and variability accounted for in confidence intervals versus prediction intervals.

Key Concepts

Predicted Value in Linear Regression

The predicted value is the estimated response (height) for a given explanatory variable (age) using the regression equation. It is calculated by plugging the specific x-value into the regression line equation, resulting in a single point estimate that does not change regardless of the interval type.

Confidence Interval for the Mean Response

A confidence interval for the mean response estimates the range in which the average height for boys of a certain age is expected to fall. This interval is narrower because it reflects uncertainty about the mean height, which is more precisely estimated than individual values.

Prediction Interval for an Individual Response

A prediction interval estimates the range where an individual boy's height is likely to fall for a given age. It is wider than the confidence interval because it accounts for both the uncertainty in the mean estimate and the natural variability among individual heights.