Using R for Introductory Statistics : John Verzani

Confidence intervals 197distributed. The sampling distribution of is asymptotically normal, as each isasymptotically normal. Consequently, the standardized statistic(7.2)will have an approximately normal distribution, with mean a and variance 1 for large n xand n y . For small n x and n y , T will have the t-distribution.The standard error of is computed differently depending on the assumptions.For independent random variables, the variance of a sum is the sum of a variance. This isused to show that the variance ofWhen the two population variances are equal, the data can be pooled to give anestimate of the common variance σ 2 . Let be the pooled estimate. It is defined by(7.3)When the population variances are not equal, the sample standard deviations are used toestimate the respective population standard deviations.The standard error is then(7.4)The statistic T will have a sampling distribution given by the t-distribution. When thevariances are equal, the sampling variation of s p is smaller, as all the data is used toestimate σ. This is reflected in a larger value of the degrees of freedom. The values usedare(7.5)(The latter value is the Welch approximation.)Given this, the T statistic is pivotal, allowing for the following confidence intervals.Confidence intervals for difference of means for two independent samplesLetbe two independent samples with distributionNormal(µ i , σ i ), i=x or y. A (1−α)· 100% confidence interval of the formcan be found where t* is given by the t-distribution. This is based on the samplingdistribution of T given in Equation 7.2.