Stat 100 Mar. 18

Stat 100 Read Ch. 18, Try 1-6,10-11 Read Ch. 19, Try 1-7

Commonplace objective Use a sample to estimate something about a population For instance, what percent of the PSU population thinks there should be 14-week semesters? Or, what is mean hours of study per week for PSU students?`

Recent Gallup Poll 34% of n=1,012 said there was an area within a mile of home where they’d be afraid to walk alone at night This is a sample estimate of a population value Reported margin of error = 3%

Margin of Error Likely upper bound on sampling error For 95% of random samples of a specific size, sampling error is less than the margin of error.

Confidence Interval An interval of values that is likely to include the true population value. Compute Sample value ± margin of error For walk alone at night question, confidence interval is 34% ± 3% This interval is likely to include percent afraid to walk alone for whole population

Confidence Level Probability that confidence interval catches the population value Confidence is actually in the procedure. To be 95% confident means that 95% of the time, the procedure provides an interval that catches the population value

95% confidence most common Most polls (including Gallup) report margin of error for 95% confidence

Another Crime Poll Results 22% said crime was an extremely serious problem in the U.S. A 95% confidence interval for the population percent who thinks this is 22%±3%

Conservative Calculation 95% margin of error = Provides value larger than actual error in at least 95% of all samples.

Examples of Margin of Error n=400, 1/sqrt(400) =0.05 or 5% n=1000, 1/sqrt(1000)=0.03, or 3% n=2500, 1/sqrt(2500)=0.02, or 2% Notice that big increase in sample size needed to drop m.e. from 3% to 2%

More Exact Margin of Error 95% m.e. =2×

For Serious Crime Question p=0.22, n=1012 Sqrt [0.22 (1-.22)/1012]=0.013 Margin of error = 2 (0.013) = Interval is 22% ± 2.6%

Crime in own neighborhood Only 3% thought crime was extremely serious in their own neighborhood. s.d.=sqrt [(0.03)(0.97)/1012]= Margin of error= 2(0.0053)=0.016, or 1.6% Confidence interval is 3% ±1.6%