Question - Margin of error for the predicted response. Refer to Example. What is the 95% margin of error of y^{^} when x = 9.0? Would you expect the 95% margin of error to be larger, smaller, or the same for x = 10.0? Explain.

Example - Prediction interval for an average of 9000 steps per day. Let's find the prediction interval for a future observation of BMI when a college-aged woman averages 9000 steps per day. The predicted value is the same as the estimate of the average BMI that we calculated in Example 10.11, that is, 23.7 kg/m2. Software tells us that the 95% prediction interval is 16.4 to 31.0 kg/m2. This interval is extremely wide, covering BMI values that are classified as underweight and obese. Because of the large amount of scatter about the regression line, prediction intervals here are relatively useless.

While a larger sample would better estimate the population regression line, it would not reduce the degree of scatter about the line. In other words, prediction intervals for BMI given activity level will always be wide. This example clearly demonstrates that a very small P-value for the significance test for a zero slope does not necessarily imply that we have found a strong predictive relationship.