Page 322 - Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 305

g(p)

p
L 1 L 2 1

Figure 9.11 Parabola defined by Equation (9.150)

Upon transposing the right-hand side, we have

u 2 =2 ! u 2 =2 ! 2
p 2 1 ‡ p 2X ‡ ‡ X 0: 9:150†
n n

In Equation (9.150), the left-hand side defines a parabola, as shown in Figure
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

8 ! " # 1=2 9 ! 1
< u 2 =2 u =2 u 2 =2 = u 2 =2
L 1;2 ˆ X ‡ X 1 X†‡ 1 ‡ : 9:151†
2n n 1=2 4n n
: ;
For large n, they can be approximated by

1=2
X 1 X†

L 1;2 ˆ X u =2 9:152†
n
An approximate [100(1 )]% confidence interval for p is thus given by, for
large n,

1=2 1=2 !

X 1 X† X 1 X†
P X u =2 < p < X ‡ u =2 ˆ 1 9:153†
n n

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