Applied Statistics and Probability for Engineers

8-5 A LARGE-SAMPLE CONFIDENCE INTERVAL FOR A POPULATION PROPORTION 267

An estimate of pis required to use Equation 8-26. If an estimate from a previous sam-

ple is available, it can be substituted for pin Equation 8-26, or perhaps a subjective estimate

can be made. If these alternatives are unsatisfactory, a preliminary sample can be taken,

computed, and then Equation 8-26 used to determine how many additional observations are

required to estimate pwith the desired accuracy. Another approach to choosing nuses the fact

that the sample size from Equation 8-26 will always be a maximum for p0.5 [that is,

p(1p)  0.25 with equality for p0.5], and this can be used to obtain an upper bound on

n. In other words, we are at least 100(1)% confident that the error in estimating pby

is less than Eif the sample size is

pˆ

pˆ

pˆ

na (8-27)

z  2

E

b

2

1 0.25 2

The approximate 100(1)% lower and upper confidence bounds are

(8-28)

respectively.

pˆz

B

pˆ 11 pˆ 2

n p^ and^ ppˆz^ B

pˆ 11 pˆ 2

n

na (8-26)

z  2

E

b

2

̨^ p 11 p 2

EXAMPLE 8-7 Consider the situation in Example 8-6. How large a sample is required if we want to be 95%

confident that the error in using to estimate pis less than 0.05? Using 0.12 as an initial

estimate of p, we find from Equation 8-26 that the required sample size is

If we wanted to be at least95% confident that our estimate of the true proportion pwas

within 0.05 regardless of the value of p, we would use Equation 8-27 to find the sample size

Notice that if we have information concerning the value of p, either from a preliminary sam-

ple or from past experience, we could use a smaller sample while maintaining both the desired

precision of estimation and the level of confidence.

One-Sided Confidence Bounds

We m ay find approximate one-sided confidence bounds on pby a simple modification of

Equation 8-25.

na

z0.025

E

b

2

1 0.25 2 a

1.96

0.05

b

2

1 0.25 2  385

pˆ

na

z0.025

E

b

2

pˆ 11 pˆ 2 a

1.96

0.05

b

2

0.12 1 0.88 2  163

pˆ pˆ

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