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Stats 3022 - Chapter 2 Inference Using t-Distributions Stat...

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Chapter 2: Inference Using t -Distributions Stat 3022 Section 001 Fall 2011 September 19, 2011 Stat 3022 (Section 001) Chapter 2 September 19, 2011 1 / 61
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Anatomical Abnormalities Associated with Schizophrenia – An Observational Study Example Question : Are any physiological indicators associated with schizophrenia? Data : 15 pairs of identical twins, one is schizophrenic, the other is not. Measured volumes (cm 3 ) of regions and subregions inside twins’ brains. Subregion of interest is the left hippocampus. What is the magnitude of the difference in volumes between unaffected and affected individuals? Can the observed difference be attributed to chance? This is an example of paired observations . Stat 3022 (Section 001) Chapter 2 September 19, 2011 2 / 61
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Data UNAFFECT AFFECTED DIFFERENCE 1 1.9 1.3 0.67 2 1.4 1.6 -0.19 3 1.6 1.5 0.09 4 1.6 1.4 0.19 5 2.1 1.9 0.13 6 1.7 1.3 0.40 7 1.8 1.7 0.04 8 1.8 1.7 0.10 9 1.8 1.3 0.50 10 1.9 1.9 0.07 11 1.2 1.0 0.23 12 1.9 1.3 0.59 13 2.0 2.0 0.02 14 1.6 1.6 0.03 15 2.1 2.0 0.11 Stat 3022 (Section 001) Chapter 2 September 19, 2011 3 / 61
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Exploratory Data Analysis > stem(DIFFERENCE, scale=2) The decimal point is 1 digit(s) to the left of the | -1 | 9 -0 | 0 | 23479 1 | 0139 2 | 3 3 | 4 | 0 5 | 09 6 | 7 > mean(DIFFERENCE); sd(DIFFERENCE) [1] 0.1986667 [1] 0.2382935 Stat 3022 (Section 001) Chapter 2 September 19, 2011 4 / 61
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Conclusions The mean difference in left hippocampus volumes between schizophrenic individuals and their nonschizophrenic twins is about 0.20, in this sample . To generalize these results to all such individuals, we must first establish the probability distribution of the sample mean . Could this observed difference have happened just by random chance? Stat 3022 (Section 001) Chapter 2 September 19, 2011 5 / 61
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Populations & Parameters Suppose there’s a population of interest that we wish to study: ex : population of undergraduate students at the University of Minnesota and a characteristic we wish to measure: ex : score on the math section of the SAT Interested in the distribution of that characteristic in the population. We summarize that distribution using parameters . Definition A parameter is an unknown numerical value describing the distribution of a random variable. Stat 3022 (Section 001) Chapter 2 September 19, 2011 6 / 61
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General Concepts & Notation Let X be a random variable ; its value fluctuates by chance ex : X = the SAT math score of a randomly selected U of M student Distribution of X can be summarized by: its mean μ (measure of central tendency) its standard deviation σ (measure of spread) Two commonly-used parameters used to describe population distributions. Particularly interested in the mean μ . Unknown in practice, must estimate it based on a random sample. Stat 3022 (Section 001) Chapter 2 September 19, 2011 7 / 61
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The Sampling Distribution of a Sample Average We estimate parameters using statistics . Let x 1 , . . . , x n be a random sample from the population distribution of X , with mean μ and standard deviation σ . The sample mean ¯ x = 1 n n X i =1 x i follows a probability distribution .
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