HP 48gII hp 48gII_user's manual_English_E_HDPMSG48E67_V2.pdf - Page 597

Confidence intervals for sums and differences of mean values If the population variances σ12 and σ22 are known, the confidence intervals for the difference and sum of the mean values of the populations, i.e., µ1±µ2, are given by: ( X 1 ± X 2 ) − zα /2 ⋅ σ 2 1 n1 + σ 2 2 n2 , (X1 ± X 2 ) + zα / 2 ⋅ σ 2 1 n1 + σ 2 2 n2 For large samples, i.e., n1 > 30 and n2 > 30, and unknown, but equal, population variances σ12 = σ22, the confidence intervals for the difference and sum of the mean values of the populations, i.e., µ1±µ2, are given by: ( X 1 ± X 2 ) − zα / 2 ⋅ S12 n1 + S 2 2 n2 , (X1 ± X2)+ zα / 2 ⋅ S12 n1 + S 2 2 n2 If one of the samples is small, i.e., n1 < 30 or n2 < 30, and with unknown, but equal, population variances σ12 = σ22, we can obtain a "pooled" estimate of the variance of µ1±µ2, as sp2 = [(n1-1)⋅s12+(n2-1)⋅s22]/( n1+n2-2). In this case, the centered confidence intervals for the sum and difference of the mean values of the populations, i.e., µ1±µ2, are given by: ( ) (X1 ± X 2 ) − tν ,α /2 ⋅ s 2 p , (X1 ± X 2) + tν ,α /2 ⋅ s 2 p where ν = n1+n2-2 is the number of degrees of freedom in the Student's t distribution. In the last two options we specify that the population variances, although unknown, must be equal. This will be the case in which the two samples are taken from the same population, or from two populations about which we suspect that they have the same population variance. However, if we have Page 18-26