Chapter 11: Comparing Two Populations or Treatments Section 11.1: Inferences Concerning the Difference Between Two Population or Treatment Means Using Independent Samples Notation Mean Value Variance Standard Deviation Population or Treatment 1 1 12 1 Population or Treatment 2 2 22 2 Sample Size Mean Variance Standard Deviation Population or Treatment 1 n1 x1 s12 s1 Population or Treatment 2 n2 x2 s 22 s2 Properties of the Sampling Distribut ion of x1 x2 If the random samples on which x1 and x 2 are based are selected independen tly of one another, then 1. x1 x2 ( mean value of x1 x2 ) x1 x2 1 2 Thus, the sampling distributi on of x1 x2 is always centered at the value of 1 2 , so x1 x2 is an unbiased statistic for estimating 1 2 . 2. 2 x1 x 2 12 22 ( variance of x1 x2 ) n1 n2 2 x1 2 x2 and x x 1 2 12 22 (standard deviation of x1 x2 ) n1 n2 3. If n1 and n 2 are both large or if the population distributi ons are (at least approximat ely) normal, then both x1 and x 2 have (at least approx) a normal distributi on. This implies that the sampling distributi on of x1 x2 is also normal or approximat ely normal. • Properties 1 and 2 follow from the following general results: 1. The mean value of a difference in means is the difference of the two individual mean values. 2. The variance of a difference of independent quantities is the sum of the two individual variances. When n1 and n2 are both large or when the population distributions are (at least approximately) normal, the distribution of z x1 x2 ( 1 2 ) 12 n1 22 n2 Is described (at least approximately) by the standard normal (z) distribution. When two random samples are independently selected and when n1 and n2 are both large or if the population distributions are normal, the standardized variable t x1 x2 ( 1 2 ) has approximately a t distribution with (V1 V2 ) df 2 2 V1 V2 n1 1 n2 1 s12 s22 n1 n2 where s12 s22 V1 and V2 n1 n2 The degrees of freedom should be rounded down to obtain an integer value. Example • Do children diagnosed with attention deficit/hyperactivity disorder (ADHD) have smaller brains than children without this condition? This question was the topic of a research study described in a paper. Brain scans were completed for 152 children with ADHD and 139 children of similar age without ADHD. Summary values for total cerebral volume (in milliliters) are given in the following table: n 152 x 1059.4 s 117.5 Children without ADHD 139 1104.5 111.3 Children with ADHD Does this data provide evidence that the mean brain volume of children with ADHD is smaller than the mean for children without ADHD? Let’s test the relevant hypotheses using a .05 level of significance. We will use the 9 steps to test the hypothesis. 1. 1 true mean brain volume for children with ADHD 2 true mean brain volume for children without ADHD 1 2 difference in mean brain volume. 2. H 0 : 1 2 0 3. H a : 1 2 0 4. Significan ce Level : .05 5. Test Statistic : t x1 x2 hypothesiz ed value 2 1 2 2 s s n1 n2 x1 x2 0 s12 s22 n1 n2 6. Assumptions: The paper states that the study controlled for age and that the participants were “recruited from the local community.” This is not equivalent to random sampling, but the authors of the paper (five of whom were doctors at well-known medical institutions) believed that it was reasonable to regard these samples as representative of the two groups under study. Both sample sizes are large, so it is reasonable to proceed with the two-sample t test. 7. Calculatio ns : (1059.4 1104.5) 0 45.10 45.10 t 3.36 2 2 13 . 415 90.831 89.120 (117.5) (111.3) 152 139 8. P - value : We first compute the number of degrees of freedom for the two - sample t test : s12 s22 V1 90.831 V2 89.120 n1 n2 (V1 V2 )2 (90.831 89.120)2 32,382.362 df 288.636 V12 V22 (90.831)2 (89.120)2 112.191 n1 1 n2 1 151 138 We round down the degrees of freedom to 288. P-value is approx. 0 because the t score is too large. 9. Conclusion: Because P-value ≈ 0 < .05, we reject H0. There is convincing evidence that the mean brain volume for children with ADHD is smaller than the mean for children without ADHD. Two-Sample t Test for Comparing Two Treatments When 1. treatments are randomly assigned to individuals or objects (or vice versa: individuals or objects are randomly assigned to treatments) 2. the sample sizes are large (in general 30 or larger) or the treatment response distributions are approximately normal The two-sample t test can be used to test H0: μ1 – μ2 = hypothesized value, where μ1 and μ2 represent the mean response for Treatments 1 and 2, respectively. In this case, these two conditions replace the assumptions previously stated for comparing two population means. The validity of the assumption of normality of the treatment response distributions can be assessed by constructing a normal probability plot or a boxplot of the response values in each sample.