Inference: Tests & Estimation for Differences in Means & Proportions, Study notes of Mathematics

Download Inference: Tests & Estimation for Differences in Means & Proportions and more Study notes Mathematics in PDF only on Docsity! Inference (hypothesis tests and estimation) Differences between means, proportions in two populations The idea: We have two populations and a variable of interest. We may be interested in the difference in mean values (on this variable) in the two populations [difference of means] or in the difference in proportions of the populations that have a particular value on the variable [difference of proportions] In dealing with means, we want to either 1. Estimate the difference between the mean values (this involves a confidence interval) or 2. Decide whether we have evidence of a difference between the mean values (this involves a test) In dealing with proportions, we want to either 1. Estimate the difference between the proportions (in the two populations) that give a certain value on the variable (confidence interval) or 2. Decide whether there is a difference between the proportions (in the two populations) that give a certain value on the variable (Test) The methods parallel the methods for estimation and for tests on the mean of one population, but the calculations are different because we have different (and more complicated) distributions. There is a special situation [the “matched samples” case] which is usually discussed with (and often confused with) inference on two populations but is really a special case of inference on one population [of differences]. Difference of means (independent samples) The basic important fact is that our best estimator of µ1−µ2 (the difference between the population means — order of subraction matters) is the difference between sample means x̄1− x̄2 . The mean of the difference in sample means (as long as we keep sample sizes the same) is exactly the difference in the population means (in the same order):µx̄1−x̄2 = µX1−µX2 and the variance of x̄1 − x̄2 is the sum of the variances of x̄1 and x̄2, so σx̄1−x̄2 = √ σ2x1 n1 + σ2X2 n2 . In addition, if X1 and X2 are approximately normally distributed or if the sample sizes are large enough then the distribution of x̄1 − x̄2 is approximately normal (which means x̄1−x̄2−(µ1−µ2)r σ2x1 n1 + σ2 X2 n2 is a Z). Thus, if we happen to know σ1 and σ2 and n1, n2 are large enough or X1, X2 are approximately normal, our 1 − α confidence interval for µ1 − µ2 is given by x̄1 − x̄2 ± E with E = Zα2 √ σ2x1 n1 + σ2X2 n2 In the usual situation, we don’t know σ1, σ2; the difference of sample means, compared using s1, s2 involves four values that vary from case to case, and is not even really a t – it is closely approximated by a t (if X1, X2 normal or n1, n2 large) but ( to make the approximation work) we have to use a strange value of degrees of freedom, so our interval for confidence 1− α is given by x̄1 − x̄2 ± E with E = tα2 √ s2x1 n1 + s2X2 n2 df = ( s21 n1 + s 2 2 n2 )2 1 n1−1 ( s21 n1 )2 + 1n2−1 ( s22 n2 )2 [This is the fractional degrees of freedom value that will be reported by your calculator or by Minitab if you use either of these for the calculation) Testing follow the same six-step procedure as testing on one mean, but slightly different numbers appear. There are the same three forms for the alternative — the order in which the two populations are identified will matter for one-sided tests. “Greater” H0 : µ1 = µ2 Ha : µ1 > µ2 or H0 : µ1 − µ2 = 0 Ha : µ1 − µ2 > 0 “Less” H0 : µ1 = µ2 Ha : µ1 < µ2 or H0 : µ1 − µ2 = 0 Ha : µ1 − µ2 < 0 “not equal” H0 : µ1 = µ2 Ha : µ1 6= µ2 or H0 : µ1 − µ2 = 0 Ha : µ1 − µ2 6= 0 Reject H0 if sample t > tα Reject H0 if sample t < −tα Reject H0 if sample t < −tα2 or sample t > tα 2 sample t = x̄1 − x̄2 − (µ1 − µ2)√ s2x1 n1 + 22X2 n2 with df = ( s21 n1 + s 2 2 n2 )2 1 n1−1 ( s21 n1 )2 + 1n2−1 ( s22 n2 )2 1