Course Notes - Department of Mathematics and Statistics

qnorm(0.975) as 2.5% of the area is in each tail.0.95 = P r(−1.96 < Z < 1.96)= P r(−1.96 < X − µ Xσ X √ n< 1.96)= P r(−1.96 σX√ n< X − µ X < 1.96 σX√ n)= P r(µ X − 1.96 σX√ n< X < µ X + 1.96 σX√ n)CONFIDENCE INTERVAL FORMULAWe are 95% confident that the unknown population mean µ X satisfiesx − 1.96 σX√ n< µ X < x + 1.96 σX√ nOrx ± 1.96 σX√ nNote that the confidence interval formula is of the form: estimate formean ± multiplier × standard error of the meanCONFIDENCE INTERVAL NOTESWe now have an interval estimate for the population mean.• A 99% confidence interval replaces the multiplier 1.96 with 2.58.Use qnorm(0.995).• Consequently the 99% C.I. is wider (less precise).• As n increases the standard error of the sample mean, √ σ Xn, getssmaller and the confidence interval is narrower (more precise) i.e.better estimate with larger n.95