Fundamentals of Probability and Statistics for Engineers

Upon transposing the right-hand side, we have

In Equation (9.150), the left-hand side defines a parabola, as shown in F igure
9.11, and two roots L 1 and L 2 of Equation (9.150) with the equal sign define the
interval within which the parabola is negative. Hence, solving the quadratic
equation defined by Equation (9.150), we have

For large n, they can be approximated by

An approximate [100(1 )]% confidence interval for p is thus given by, for
large n,

1

p

L 1 L 2

g(p)

Figure 9.11 Parabola defined by Equation (9.150)

Parameter Estimation 305

p^21 ‡

u^2 = 2

n

!

p 2 X‡

u^2 = 2

n

!

‡X

2

 0 : … 9 : 150 †

L 1 ; 2 ˆ X‡

u^2 = 2

2 n

!



u= 2

n^1 =^2



X… 1 X†‡

u^2 = 2

4 n

(^8) "# 1 = 2
<
:

9

=

;

1 ‡

u^2 = 2

n

! 1

: … 9 : 151 †

L 1 ; 2 ˆXu= 2

X… 1 X†

n

 1 = 2

… 9 : 152 †



P Xu= 2

X… 1 X†

n

 1 = 2

<p<X‡u= 2

X… 1 X†

n

 1 = 2

ˆ 1  … 9 : 153 †

!