Prediction interval for the average response of multiple new observations

Consider the normal simple linear regression model y; = n + E13.- + 5., 13 = 1,. .. ,n with a; m N{D,cr). Denote ha, in and Er as the least square estimates of al and o. In the class, we established the prediction intenral of the response y for one new observation 135. Now, suppose we have M new observations at 235, so their corresponding responses are denoted as pal, . . . , ydm- Instead of predicting the response corresponding to each observation separately, we consider the average of new responses _* _ 361+\" sEM HI] _ M ' Derive a ll][1 :1} (H: prediction interval for 175 (Hint: starting with the point estimate and its variance, and one wag,r to check your answer is when M = 1, you should get eivzactl},r the same prediction interval derived in the lecture)