Page 367 - Fundamentals of Probability and Statistics for Engineers

350 Fundamentals of Probability and Statistics for Engineers
. Result 2: a [100(1
)% confidence interval for is determined by [see
Equation (9.141)]

8 9 1=2
1
n
" #
< X =
^
c 2
L 1; 2 ˆ B t n 2;
=2 x i x† 2 : 11:40†
iˆ1
: ;
f
. Result 3: a [100(1
)]% confidence interval for E Y gˆ ‡ x is deter-
mined by [see Equation (9.141)]
8 8 99 1=2
" # 1
n
1 2 X 2
< < ==
c 2
L 1;2 ˆ EfYg t n 2;
=2 ‡ x x† x i x† : 11:41†
d
n
iˆ1
: : ;;
. Result 4: a two-sided [100(1
)% confidence interval for 2 is determined
by [see Equation (9.144)]
n 2†
c 2
L 1 ˆ ;
2
n 2;
=2
11:42†
n 2†
c 2
L 2 ˆ :
2
n 2;1
=2
If a one-sided confidence interval for 2 is desired, it is given by [see Equation
(9.145)]
c 2
n 2†
L 1 ˆ : 11:43†
2
n 1;
A number of observations can be made regarding these confidence intervals. In
each case, both the position and the width of the interval will vary from sample
to sample. In addition, the confidence interval for ‡ x is shown to be a
function of x. If one plots the observed values of L 1 and L 2 they form a
confidence band about the estimated regression line, as shown in Figure 11.4.
Equation (11.41) clearly shows that the narrowest point of the band occurs at
x ˆ x; it becomes broader as x moves away from x in either direction.

Example 11.4. Problem: in Example 11.3, assuming that Y is normally

distributed, determine a 95% confidence band for ‡ x.

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