There, as the number of experiments increased, the distribution narrowed and the confidence interval became tighter around the expected value of the mean. The logic here is similar, although not identical, to that discussed when developing the relationship between the sample size and the confidence interval using the Central Limit Theorem. The expected value method assumes that the experiment is conducted multiple times rather than just once as in the other method. The confidence interval, measuring the expected value of the dependent variable, is smaller than the prediction interval for the same level of confidence. The test statistics for these two interval measures within which the estimated value of y will fall are:įigure 13.15 Prediction and confidence intervals for regression equation 95% confidence level.įigure 13.15 shows visually the difference the standard deviation makes in the size of the estimated intervals. The second case, where we are asking for the estimate of the impact on the dependent variable y of a single experiment using a value of x, is called the prediction interval. To avoid confusion, the first case where we are asking for the expected value of the mean of the estimated y, is called a confidence interval as we have named this concept before. Both are correct answers to the question being asked, but there are two different questions.
The conclusion is that we have two different ways to predict the effect of values of the independent variable(s) on the dependent variable and thus we have two different intervals. Because this approach acts as if there were a single experiment the variance that exists in the parameter estimate is larger than the variance associated with the expected value approach. The second approach to estimate the effect of a specific value of x on y treats the event as a single experiment: you choose x and multiply it times the coefficient and that provides a single estimate of y. Remember that there is a variance around the estimated parameter of x and thus each experiment will result in a bit of a different estimate of the predicted value of y. Here the question is: what is the mean impact on y that would result from multiple hypothetical experiments on y at this specific value of x. The first approach wishes to measure the expected mean value of y from a specific change in the value of x: this specific value implies the expected value. There are actually two different approaches to the issue of developing estimates of changes in the independent variable, or variables, on the dependent variable. This was why we developed confidence intervals for the mean and proportion earlier. Remember that point estimates do not carry a particular level of probability, or level of confidence, because points have no “width” above which there is an area to measure.
Ŷ = b 0 + b, X 1 i + ⋯ + b k X k i ŷ = b 0 + b, X 1 i + ⋯ + b k X k i