The prediction interval is calculated in a similar way using the prediction standard error of 8.24 (found in cell J12). The confidence interval, calculated using the standard error of 2.06 (found in cell E12), is (68.70, 77.61). Referring to Figure 2, we see that the forecasted value for 20 cigarettes is given by FORECAST(20,B4:B18,A4:A18) = 73.16. ExampleĮxample 1: Find the 95% confidence and prediction intervals for the forecasted life expectancy for men who smoke 20 cigarettes in Example 1 of Method of Least Squares.įigure 2 – Confidence and prediction intervals Where the standard error of the prediction isįor any specific value x 0 the prediction interval is more meaningful than the confidence interval. The 95% prediction interval of the forecasted value ŷ 0 for x 0 is Again, this is not quite accurate, but it will do for now. Here we look at any specific value of x, x 0, and find an interval around the predicted value ŷ 0 for x 0 such that there is a 95% probability that the real value of y (in the population) corresponding to x 0 is within this interval (see the graph on the right side of Figure 1). There is also a concept called a prediction interval. any of the lines in the figure on the right above). Thus there is a 95% probability that the true best-fit line for the population lies within the confidence interval (e.g. The confidence interval consists of the space between the two curves (dotted lines). In the graph on the left of Figure 1, a linear regression line is calculated to fit the sample data points. This is not quite accurate, as explained in Confidence Interval, but it will do for now.įigure 1 – Confidence vs. The 95% confidence interval is commonly interpreted as there is a 95% probability that the true linear regression line of the population will lie within the confidence interval of the regression line calculated from the sample data.
How to calculate these values is described in Example 1, below. Here, s y⋅ x is the standard estimate of the error, as defined in Definition 3 of Regression Analysis, S x is the squared deviation of the x-values in the sample (see Measures of Variability), and t crit is the critical value of the t distribution for the specified significance level α divided by 2. The 95% confidence interval for the forecasted values ŷ of x is In Confidence and Prediction Intervals we extend these concepts to multiple linear regression, where there may be more than one independent variable. We also show how to calculate these intervals in Excel. a linear regression with one independent variable x (and dependent variable y), based on sample data of the form ( x 1, y 1), …, ( x n, y n).
On this webpage, we explore the concepts of a confidence interval and prediction interval associated with simple linear regression, i.e.
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