12 Consider the following sample of fat content (in percentage) of n 5 10 randomly

Example 7.12 Consider the following sample of fat content (in percentage) of n 5 10 randomly

  selected hot dogs (“Sensory and Mechanical Assessment of the Quality of Frankfurters,” J. of Texture Studies, 1990: 395–409):

  Assuming that these were selected from a normal population distribution, a 95 CI for (interval estimate of) the population mean fat content is

  x6t 4.134

  s

  Suppose, however, you are going to eat a single hot dog of this type and want a pre- diction for the resulting fat content. A point prediction, analogous to a point esti- mate, is just x 5 21.90 . This prediction unfortunately gives no information about reliability or precision.

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  CHAPTER 7 Statistical Intervals Based on a Single Sample

  The general setup is as follows: We have available a random sample

  X 1 ,X 2 , c, X n from a normal population distribution, and wish to predict the value of n11 , a single future observation (e.g., the lifetime of a single lightbulb to be purchased

  or the fuel efficiency of a single vehicle to be rented). A point predictor is , and the X resulting prediction error is X2X n11 . The expected value of the prediction error is

  E(X 2 X n11 ) 5 E(X) 2 E(X n11 )5m2m50

  Since X n11 is independent of 1 , c, X n , it is independent of , so the variance of X

  the prediction error is

  s 2 2 2 1

  V(X 2 X n11 ) 5 V(X) 1 V(X n11 )5 n 1s 5s a1 1 n b

  The prediction error is a linear combination of independent, normally distributed rv’s, so itself is normally distributed. Thus

  has a standard normal distribution. It can be shown that replacing s by the sample

  standard deviation S (of X 1 , c, X n ) results in X2X

  T5

  n11

  | t distribution with n 2 1 df

  1 S 11 É n

  Manipulating this T variable as T 5 (X 2 m)(S2n) was manipulated in the devel- opment of a CI gives the following result.

  PROPOSITION

  A prediction interval (PI) for a single observation to be selected from a nor- mal population distribution is

  x 6 t 1

  a2,n21

  sB1 1 (7.16)

  n

  The prediction level is 100(1 2 a) . A lower prediction bound results from

  replacing by t a2 t a and discarding the 1 part of (7.16); a similar modifica-

  tion gives an upper prediction bound. The interpretation of a 95 prediction level is similar to that of a 95 confidence

  level; if the interval (7.16) is calculated for sample after sample, in the long run 95 of these intervals will include the corresponding future values of X.

  Example 7.13 With n 5 10, x 5 21.90, s 5 4.134 , and t .025,9 5 2.262 , a 95 PI for the fat content (Example 7.12

  of a single hot dog is

  This interval is quite wide, indicating substantial uncertainty about fat content. Notice that the width of the PI is more than three times that of the CI.

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  7.3 Intervals Based on a Normal Population Distribution

  The error of prediction is X2X n11 , a difference between two random variables, whereas the estimation error is X2m , the difference between a random variable and a fixed (but unknown) value. The PI is wider than the CI because there is more variability

  in the prediction error (due to X n11 ) than in the estimation error. In fact, as n gets arbi-

  trarily large, the CI shrinks to the single value m, and the PI approaches

  m6z

  a2 s.

  There is uncertainty about a single X value even when there is no need to estimate.