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Minimum Sample Size for Confidence Interval

```
Date: 09/21/2000 at 11:31:54
From: Steve
Subject: Sample size for estimate for population proportion

Let's say I have a population of 400 people. I need to find out if a
value for each person is 1 or 0. Sampling each person is very time-
consuming, however, so I don't want to have to sample any more than
necessary.

I need to find out how many people I need to sample to say that the
proportion correct (1 = correct) in the sample is representative of
the proportion correct in the population. I am trying to make a case
that if the randomly drawn sample I choose is done right, I am
"whatever" amount confident that the entire population is correct at
the same rate.

Thanks,
Steve
```

```
Date: 09/21/2000 at 15:55:56
From: Doctor Anthony
Subject: Re: Sample size for estimate for population proportion

Since we are talking about the means of samples we can by the Central
Limit theorem use a normal distribution.

If p = proportion of 1's, then if we take a random sample of size n,
and ps is the unbiased estimator of p we get (using the normal
approximation to the binomial),

ps - p
z = -------------
sqrt(ps.qs/n)

where qs = 1 - ps.

With 90% confidence limits we have:

ps-p
Prob[-1.645 < ------------- < 1.645] = 0.90
sqrt(ps.qs/n)

Prob[ps-1.645*sqrt(ps.qs/n) < p < ps+1.645*sqrt(ps.qs/n)] = 0.90

Now if we are to have a margin of error of not more than say 10% we
require:

ps + 1.645 sqrt(ps.qs/n) = ps + 0.1
so
1.645 sqrt(ps.qs/n) = 0.1

since we don't know the value of ps, we must assume the worst case
with ps.qs a maximum. That is:

ps(1 - ps) = maximum

ps - ps^2 = maximum

Differentiating:

1 - 2.ps = 0
so
ps = 1/2   and   qs = 1 - ps = 1/2

So we can put

ps.qs = 0.5 x 0.5 = 0.25

In the worst case

1.645*sqrt(0.25/n) = 0.1

squaring

2.706 (0.25/n) = 0.01

n =  2.706 x 0.25/0.01

=  67.65

So take a sample of size 67.65 to get a 90% confidence limit with a
maximum of 10% margin of error.

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Statistics

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