Confidence Intervals Mon, March 22 nd

Point & Interval Estimates  Point estimate – use sample to estimate exact statistic to represent pop parameter –Point estimate of average Amer salary = $29, 340  Interval estimate – use sample to estimate a range of values within which pop parameter may fall (Confidence Interval) –Interval estimate of average salary = $27,869 to $30,811

(cont.)  With confidence interval, specify likelihood this interval will contain the pop parameter –95% conf interval, means we are 95% confident the interval/range contains the true pop parameter –Almost always choose 90, 95, or 99% confidence

Constructing Confidence Interval  1) Calculate standard error of the mean  ybar =  y / sqrt N  2) Decide on confidence level (90/95/99) – then find corresponding z value –We know that, for a normal curve, 68% of the scores will fall betw + or – 1SD (std error), so –95% will fall betw + or – 1.96 SE (see normal curve table for.05 / 2 tails, so z = + or –1.96 –99% will fall betw + or –2.58 SE (see normal curve table for.01 / 2 tails, so z = + or – 2.58)

(cont.)  3) Use Conf Interval formula: CI = Ybar + and – Z(  ybar )  4) Interpret results Ex) Find 95% CI for average commuting time when ybar = 7.5 hrs,  y = 1.5 and sample N=500 *Find standard error,  ybar = 1.5 / sqrt(500) =.07

example For 95% CI, z value is 1.96 (see table 12.1 for z values for 90/95/99% CI) 95% CI = and – 1.96(.07) = 7.36 to 7.64 (7.36, 7.64) Interpretation – we are 95% confident the true commuting time of the pop is between 7.36 and 7.64 hrs per week)

Example (cont.)  Notice what happens to CI when we increase confidence to 99%  Corresponding z for 99% = 2.58, so  99% CI = and – 2.58(.07) = 7.32 to 7.68  Now only 1% risk we are wrong, but a wider, less precise, interval

Estimating  ybar  If not given  y and only given Sy (sample std dev), can estimate S ybar (rather than  ybar )  S ybar = Sy / sqrt N

Sample Size and CI  Increase N and increase precision of CI (range becomes smaller): –Due to smaller standard error –Earlier example, increase N from 500 to 2500,  ybar = 1.5 / sqrt(500) =.07  ybar = 1.5 / sqrt(2500) =.03 CI = and – 1.96(.03) = 7.44 to 7.51 Compared w/7.36 to 7.64 (w/.07 std error)