Course Notes - Department of Mathematics and Statistics

6.2.2 The t DistributionTHE t DISTRIBUTION• Often the true standard deviation σ X is not known.• We estimate with the sample standard deviation s X .• This gives larger values than 1.96 and 2.58 and hence wider, lessprecise confidence intervals.• The 95% C.I. becomes:x ± t (α2 ,ν) s X√ nwhere ν = n − 1 (degrees of freedom) and α is the combinedprobability of the two tails (here α=1-0.95=0.05).Finding the t multiplierR-commander - Distributions > Continuous distributions > tdistribution > t quantiles and enter the probability ( α 2), the degreesof freedom (ν) and choose upper tail.EXAMPLE CONTINUEDNow suppose that the pharmacologist did not know the value ofσ X and was forced to take the sample standard deviation from thesample of size n = 8 as the best estimate of σ X , namely s X = 1.5hours. Find 95% and 99% confidence intervals for µ X .The 95% Confidence IntervalThe 95% confidence interval is:Alternatively we write:x ± t (0.025,7)s X√ n= 8.4 ± 2.365 × 1.5 √8= 8.4 ± 1.25= (7.15, 9.65)7.15 < µ X < 9.6598