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Test on Pairs of Means – Case I
Suppose are iid independent of that are iid Further, suppose that n1 and n2 are large or that are known. We are interested in testing H0: μx = μy versus a one sided or a two sided alternative… Then,… STA248 week 11
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Test on Pairs of Means – Case II
Suppose are iid independent of that are iid Further, suppose that n1 and n2 are small and that are unknown but we assume they are equal to σ2. We are interested in testing H0: μx - μy = δ versus a one sided or a two sided alternative… Then,… STA248 week 11
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Example The strength of concrete depends, to some extent, on the method used for drying it. Two drying methods were tested on independently specimens yielding the following results… We can assume that the strength of concrete using each of these methods follows a normal distribution with the same variance. Do the methods appear to produce concrete with different mean strength? Use α = 0.05. STA248 week 11
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R Example The file sparrow contains data on the length of humerus in adult male sparrows who survived after a storm and sparrows who perished after the storm. Are the mean length of humerus different between these two groups of saparrows? STA248 week 11
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Test on Pairs of Means – Case III
Suppose are iid with E(Xi ) = µx and Var(Xi) = σx independent of that are iid with E(Yi ) = µy and Var(Yi) = σy Further, suppose that n1 and n2 are both large. We are interested in testing H0: μx - μy = δ versus a one sided or a two sided alternative… Then,… STA248 week 11
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Test on Two Proportions
Suppose are iid Bernoulli(θ1) independent of that are iid Bernoulli(θ2). Further, suppose that n1 and n2 are large. We are interested in testing H0: θ1 = θ2 versus a one sided or a two sided alternative… Then,… STA248 week 11
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Example STA248 week 11
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Paired Observations In a matched pairs study, subjects are matched in pairs and the outcomes are compared within each matched pair. The experimenter can toss a coin to assign two treatment to the two subjects in each pair. One situation calling for match pairs is when observations are taken on the same subjects, under different conditions. A match pairs analysis is needed when there are two measurements or observations on each individual and we want to examine the difference. This corresponds to the case where the samples are not independent. For each individual (pair), we find the difference d between the measurements from that pair. Then we treat the di as one sample and use the one sample t test and confidence interval to estimate and test the difference between the treatments effect. STA248 week 11
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Example Seneca College offers summer courses in English. A group of 20 students were given the TOFEL test before the course and after the course. The results are summarized in the next slide. Find a 95% CI for the average improvement in the TOFEL score. Test whether attending the course improve the performances on the TOFEL. STA248 week 11
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Row Student Pretest Posttest improvement 1 1 30 29 -1 2 2 28 30 2
Data Display Row Student Pretest Posttest improvement STA248 week 11
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R Example Are there physiological indicators of schizofrenia?
To answer this questions researchers collected information about 15 pairs of identical twins, one who was schizophrenic and the other one healthy. They collected measurements about their brain volume by measuring their left hippocampus. One of the research questions was if the means of left hippocampus volume is the same for schizophrenic and non-schizophrenic . STA248 week 11
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Test on Pairs of Variances
Suppose are iid independent of that are iid We are interested in testing versus a one sided or a two sided alternative… Then… STA248 week 11
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Example STA248 week 11
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