module 17 wrap up estimating a population proportion let's summarize the sample proportion is a point estimate for the population proportion so remember that sample proportion right is uh signified by P hat and population proportion signified by P but sample proportions vary we therefore use an interval estimate call the confidence interval to give us a range of values for the population proportion a range of values we can calculate a confidence interval for a population proportion when we can use a normal distribution to model the long run behavior of sample proportions we can use a normal distribution model when the sample is randomly selected and there are at least 10 observed successes and 10 observed failures in the sample the interpretation of a confidence interval depends on the confidence level for example using a 95% confidence level we would say that we are 95% confident that the population proportion Falls within the interval 95% confidence confident means that in the long run 95% of random samples will generate intervals that contain the population proportion and 5% will not in other words 5% of the time will get a proportion that's actually not telling the truth about the population proportion so we'll get a confidence interval that's not going to capture the true population proportion a confidence in is a sample proportion so P hat plus or minus a margin of error so margin of error the margin of error is related to the confidence level for a 95% confidence level the margin of error is approximately two centered errors so the formula is we have P hat Plus plus or minus and this part is the margin of error and remember that um this Z with a subc in it that's what's related to the confidence um level so that's a zcore related to the confidence level so for example if you're doing a 95% confidence interval then that Z value is going to be 1.645 but if you're doing instead a 95% create a 95% confidence interval then you're going to use 1.96 for that value for Z and if you're creating a 99% confidence interval you're going to use 2.58 for that zcore now lower confidence levels and higher sample sizes so lower confidence uh levels and higher sample sizes lead to narrower confidence intervals a narrower confidence interval has a smaller error since we want to be confident that an interval accurately estimates the population proportion high levels of confidence are desirable so larger sample sizes are the preferred way to decrease the error and create narrower comp confidence intervals