S??(Xi?X)2/n. Find the constant c so that the statistic
21ncX?Xn?1?has a t-distribution. If n=8, determine k such
thatPr(X?kS???X?kS)?0.80. The observed interval is(x?ks,x?ks) often called an 80 percent prediction interval forX9. Solution: X~N(?,?2/n)，Xn?1~N(?,?2)，X?Xn?1~N(0,X?Xn?1X?Xn?1n?1?n
n?12?), n~N(0,1).nS2?2~?2(n?1),n?1?nnS2?2(n?1)?X?Xn?1n?1n??~t(n?1).nn?1STherefore c?n?1. n?1Pr(X?kS?x9?X?kS)?Pr(X?X97X?X97?k)?Pr(?k)?0.80 S9S97k?1.415,k?1.60. So k=1.60. 9
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