Question 1202937: A psychologist is interested in constructing a 99% confidence interval for the proportion of people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain. 54 of the 795 randomly selected people who were surveyed agreed with this theory. Round answers to 4 decimal places where possible.
With 99% confidence the proportion of all people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain is between
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! p = 54/795 = .0679
q = 1-p = 1-.0679 = .9321
n = 795
s = sqrt(p * q / n) = sqrt(.0679 * .9321 / 795) = .0089
at 99% two tailed confidence interval, the critical z-score will be z = plus or minus 2.5758.
when z = -2.5758, z = (x - .0679) / .0089.
solve for x to get x = .0450.
when z = 2.5758, z = (x - .0679) / .0089.
solve for x to get x = .0908.
your 99% confidence interval gets you a proportion from x = .0450 to x = .0908.
that's the 99% confidence interval of the proportion of all people who accept the theory that a person's spirit is no more than the complicted network of neurons in the brain.
since the standard error was taken from the sample p rather than from the population p, it is possible that the use of the t-score was indicated rather than the use of the z-score.
in that case, the critical t-score with 794 degrees of freedom would have been t = plus or minus 2.5820 and you would have gotten 99% confidence interval from x = .0449 to x = .0909.
the difference would be very small because the sample size was quite large.
in most cases of this type, i've seen z being used, so a good bet would be to assume z is used whether or not the standrd error is taken from the assumed population p or from the sample p.
p is the proportion of all people who accept the theory that a person's spirit is no more than the complicted network of neurons in the brain.
q is the roportion of all people who don't belie that.
n is the sample size
dof = degrees of freedom for use in t-score analysis = n - 1.
s = standard error = sqrt(p * q / n)
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