1. Introduction In this lecture we will derive the formulas for the - PDF document

Lecture 32: The prediction interval formulas for the next observation from a normal distribution when σ is unknown 0/ 5

1. Introduction In this lecture we will derive the formulas for the symmetric two-sided prediction interval for the n + 1-st observation and the upper-tailed prediction interval for the n + 1-st observation from a normal distribution when the variance σ 2 is unknown . We will need the following theorem from probability theory that gives the distribution of the statistic X − X n + 1 . Suppose that X 1 , X 2 , . . . , X n , X n + 1 is a random sample from a normal distribution with mean µ and variance σ 2 . Theorem 1 �   n + 1 � �   The random variable T = X − X n + 1 / S  has t distribution with n − 1       n  degrees of freedom. 2. The two-sided prediction interval formula Now we can prove the theorem from statistics giving the required prediction interval for the next observation xn+1 in terms of n observations x 1 , x 2 , . . . , x n . Note that it is symmetric around X . This is one of the basic theorems that you have to learn how to prove. There are also asymmetric two-sided prediction intervals. Theorem 2 � �   n + 1 n + 1   The random interval  X − t α / S , X + t α / S  is a    2 , n − 1 2 , n − 1    n n 100 ( 1 − α )% -prediction interval for x n + 1 . 1/ 5 Lecture 32: The prediction interval formulas for the next observation from a normal distribution when σ is unknown

Proof. We are required to prove   �   n + 1 √     P  X n + 1 ∈    X − t α / n + 1 nS , X + t α / S    = 1 − α.     2 , n − 1 2 , n − 1       n    We have  � �  n + 1 n + 1   LHS = P S < X n + 1 , X n + 1 < X + t α /  X − t α /  S    2 , n − 1 2 , n − 1    n n   � � = P X − X n + 1 < t α / 2 , n − 1  � �  n + 1 n + 1   = P   X − X n + 1 < t α / S , X − X n + 1 > − t α / S    2 , n − 1 2 , n − 1    n n    � �  n + 1 n + 1 � �   = P  X − X n + 1 / S < t α / 2 , n − 1 , ( X − X n + 1 ) / S > − t α /    2 , n − 1     n n   = P ( T < t α / 2 , n − 1 ) = P ( − t α / 2 , n − 1 ) = 1 − α 2 , n − 1 , T > − t α / 2 , n − 1 < T < t α / To prove the last equality draw a picture. � 2/ 5 Lecture 32: The prediction interval formulas for the next observation from a normal distribution when σ is unknown

Once we have an actual sample x 1 , x 2 , . . . , x n we obtain the the observed value � �   n + 1 n + 1   s , x + t α /  x − t α / s  for the prediction (random) interval    2 , n − 1 2 , n − 1    n n � �   n + 1 n + 1   S , X + t α /  X − t α / S  The observed value of the    2 , n − 1 2 , n − 1    n n prediction (random) interval is also called the two-sided 100 ( 1 − α )% prediction interval for x n + 1 . 3. The upper-tailed prediction interval In this section we will give the formula for the upper-tailed prediction interval for the next observation x n + 1 . Theorem 3 �   n + 1    is a 100 ( 1 − α )% -prediction The random interval  X − t α, n − 1 S , ∞       n interval for the next observation x n + 1 . Proof We are required to prove   �   n + 1     P   X n + 1 ∈   X − t α, n − 1 S , ∞   = 1 − α.            n    3/ 5 Lecture 32: The prediction interval formulas for the next observation from a normal distribution when σ is unknown

Proof (Cont.) We have  �  n + 1   LHS = P  X − t α, n − 1  S < X n + 1       n    �  n + 1   = P  X − X n + 1 < t α, n − 1  S       n   �   n + 1   = P  ( X − X n + 1 ) /  S < t α, n − 1       n   = P ( T < t α, n − 1 ) = 1 − α To prove the last equality draw a picture - I want you to draw the picture on tests and the final. � Once we have an actual sample x 1 , x 2 , . . . , x n we obtain the observed value �   n + 1    x − t α, n − 1 s , ∞  of the upper-tailed prediction (random) interval       n �   n + 1    X − t α, n − 1 S , ∞  The observed value of the upper-tailed prediction       n (random) interval is also called the upper-tailed 100 ( 1 − α )% prediction interval for x n + 1 . 4/ 5 Lecture 32: The prediction interval formulas for the next observation from a normal distribution when σ is unknown

� n + 1 The number random variable X − t α, n − 1 S or its observed value n � n + 1 x − t α, n − 1 s is often called a prediction lower bound for x n + 1 because n  �  n + 1    = 1 − α. P  X − t α, n − 1  S < X n + 1       n  5/ 5 Lecture 32: The prediction interval formulas for the next observation from a normal distribution when σ is unknown