STA291 Statistical Methods Lecture 17

Bias versus Efficiency 2 AB CD

Last time: Thanks to the CLT … We know is approximately normal with mean and standard deviation ___ and ______, respectively. Time to use this fact to do some inference about … ? 3

Estimation of a Proportion 4 The sample proportion is an unbiased and efficient point estimator of the population proportion p

Confidence Interval for a Proportion 5 o A large sample confidence interval for the population proportion p has the form o where is the sample proportion

Example 6 o In a recent telephone survey (conducted in mid- October), people were asked whether they have seen a ghost or felt its presence. o Of 1013 adults interviewed, 230 answered yes, and 783 answered no. o Find the point estimate of the population proportion of adults who would answer yes o Construct and interpret a 95% confidence interval for the population proportion. o Can you conclude that fewer than half of the population has seen a ghost or felt its presence?

Conditions & Interpretation 7 o Conditions: o Random sample o “Infinite population”, or population size at least 10 times that of the sample o p not too near 0, 1 (usually checked by verifying that &.) o Interpretation “We found the interval [ __, __ ] using a method that, if done with many randomly drawn samples, would result in intervals that would include the true population proportion (of _____) ___ % of the time.”

Sample size considerations 8 o Suppose you’re working for a candidate and you return with a 95% CI for the proportion of likely voters who replied in the affirmative when asked if they were going to vote for your candidate; that interval is [0.45,0.57]. o Cause for celebration? or consternation? o Difficulty: while the point estimate (0.51) is on the “right side” of 50%, the margin of error is so large, we aren’t confident we’re going to win. o Solution?

Sample size calculation 9 o Suppose we’re given a target bound on our margin of error, ME o This can be solved for the sample size, n: o But wait, we don’t know p…

Looking back o Estimation of proportions o Point estimate (little used) o Confidence interval estimate o Assumptions o Interpretation (!) o Sample size calculation Cases