Question 1163081: The estimate of the population proportion is to be within plus or minus 0.05, with a 90% level of confidence. The best estimate of the population proportion is 0.31. How large a sample is required?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! p = .31
q = .69
at 90% confidence level, critical z-score = plus or minus 1.645
s = sqrt(p*q/n) = sqrt(.2139/n)
m = p = .31
z-score formula is:
z = sqrt(x-m)/s
(x-m) is equal to plus or minus .05.
to find n, we work with plus 1.645 z-score and plus (x-m) = .05
z-score formula becomes:
1.645 = .05 / sqrt(.2139/n)
multiply both sides of this equation by sqrt(.2139/n) and divide both sides of this equation by 1.645 to get:
sqrt(.2139/n) = .05/1.645
square both sides of this equation to get:
.2139/n = (.05/1.645)^2
solve for n to get:
n = .2139/(.05/1.645)^2 = 231.527499
that should be your answer after you round it to the next higher integer.
for now, as is.
when n = 231.527499, s = sqrt(.2139/that) = .0303951368
your z-score formula becomes:
1.645 = (x - .31) / .0303951368
solve for x to get:
x = 1.645 * .03039513368 + .31 = .36
.36 is .05 units away from .31, so you did good.
if you solve for the low side x, the formula becomes:
-1.645 * .03039513368 + .31 = .26
your confidence level is x = .26 to .36.
the critical raw scores are plus or minus .05 units from the mean.
this is what you wanted.
your solution is that the sample size required to get exactly plus or minus .05 units from the mean is 231.527499.
since the sample size can't be a fraction, you would normally round this to the next highest integer to get a minimum sample size of 232.
that would ensure the answer was less than or equal to .05 units from the mean.
in a proportion type of problem, .....
p is the probability of success
q is the probability of failure = 1 - p
m is the mean = p
s is the standard error (not the standard deviation of the population and not the standard deviation of the sample).
the standard error is the standard deviation of the distribution of sample means.
the formula for s is:
s = sqrt(p*q/n)
x is the raw score which, in this case is the raw proportion.
i'll ve available to answer any questions you might have about this.
in the meantime, your anser is that the minimum sample size needs to be equal to 232.
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