Mathematics
The Sample Proportion

### Question Description

Sample size n is 450.

Standard deviation of a proportion p is given by sqrt(p(1-p))

sd = sqrt(0.07 x 0.93)

= 0.255

So the standard error for a sample mean is given by sd/sqrt(n) where n is the sample size:

SEM = sd/sqrt(n)

= 0.255/sqrt(450)

= 0.012

25/450 and 35/450 are 0.055555556 and 0.077777778. The probability our mean will lie between these will be given by the cdf at 0.0777778, minus the cdf at 0.0555556 (consider what the cdf means: The area under the curve to the left of that point).

So the probability that fewer than 25-25 will have no home phone in our sample of 450 is the normal cdf at .0777778 with mean = .07 and sd = SEM = 0.012 minus the normal cdf at .0555556 with mean = .07 and sd = SEM = 0.012:

= Norm.cdf (0.078, 0.07, 0.012) - Norm.cdf (0.056, 0.07, 0.012)

0.741071724 - 0.114889762

= 0.626

iainharlow (1041)
Duke University
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