Statistics problem...? A survey of 865 voters in one state reveals that 408 favor approval of an issue before the legislature. Construct the 95% confidence interval for the true proportion of all voters in the state who favor approval.

large sample confidence intervals are used to find a region in which we are 100 (1-α)% confident the true value of the parameter is in the interval.

For large sample confidence intervals for the proportion in this situation you have:

pHat ± z * sqrt( (pHat * (1-pHat)) / n)

where pHat is the sample proportion
z is the zscore for having α% of the data in the tails, i.e., P( |Z| > z) = α
n is the sample size

For a 95% confidence interval you have a z-score of z = 1.96

phat = 408/865
n = 865

the confidence interval is:

408/865 ± 1.96 * sqrt( 408/865 * (1 - 408/865) / 865)

( 0.4384088 , 0.5049438 )