The same value,
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= b0 + b1 x |
is used both to estimate the mean response at x and to predict a new individual's response at x, but the errors are different in the two situations — they tend to be larger for predicting a new value.
95% confidence interval for mean response
A formula for the standard error on the right exists, but you should rely on statistical software to find its value.
95% prediction interval for a new individual's response
For prediction, a similar interval is used:
where k is greater than the corresponding standard error for the confidence interval. Statistical software should again be used to find its value.
Example
The diagram below shows 95% confidence intervals for the mean response at x and 95% prediction intervals for a new response at x as bands for a small data set with n = 7 values.
Extrapolation
These 95% confidence intervals and 95% prediction intervals are valid within the range of x-values about which we have collected data, but they should not be relied on for extrapolation. Both intervals assume that the normal linear model describes the process, but we have no information about linearity beyond the x-values that have been collected.