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# lecture6 - MIT OpenCourseWare http/ocw.mit.edu 2.830J...

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MIT OpenCourseWare ____________ http://ocw.mit.edu 2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit: ________________ http://ocw.mit.edu/terms .

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1 M anufacturing Control of Manufacturing Processes Subject 2.830/6.780/ESD.63 Spring 2008 Lecture #6 Sampling Distributions and Statistical Hypotheses February 26, 2008
2 M anufacturing Statistics The field of statistics is about reasoning in the face of uncertainty, based on evidence from observed data Beliefs: Probability Distribution or Probabilistic model form Distribution/model parameters Evidence: Finite set of observations or data drawn from a population (experimental measurements/observations) Models: Seek to explain data wrt a model of their probability

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3 M anufacturing Topics Sampling Distributions ( χ 2 and Student’s-t) – Uncertainty of Parameter Estimates – Effect of Sample Size – Examples of Inference Inferences from Distributions – Statistical Hypothesis Testing – Confidence Intervals Hypothesis Testing The Shewhart Hypothesis and Basic SPC – Test statistics - xbar and S
4 M anufacturing Sampling to Determine Parent Probability Distribution Assume Process Under Study has a Parent Distribution p(x) Take “ n ” Samples From the Process Output ( x i ) Look at Sample Statistics (e.g. sample mean and sample variance) Relationship to Parent Both are Random Variables Both Have Their Own Probability Distributions Inferences about Process via Inferences about the Parent Distribution

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5 M anufacturing Moments of the Population vs. Sample Statistics Mean Variance Standard Deviation Covariance Correlation Coefficient Underlying model or Sample Statistics Population Probability
6 M anufacturing Sampling and Estimation • Sampling: act of making observations from populations • Random sampling: when each observation is identically and independently distributed (IID) • Statistic: a function of sample data; a value that can be computed from data (contains no unknowns) – Average, median, standard deviation – Statistics are by definition also random variables

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7 M anufacturing Population vs. Sampling Distribution Population (“true”)probability density function) Sample Mean (statistic) n = 20 n = 10 n = 2 Sample Mean Distribution (sampling distribution) n = 1
8 M anufacturing Sampling and Estimation, cont. A statistic is a random variable, which itself has a sampling (probability) distribution I.e., if we take multiple random samples, the value for the statistic will be different for each set of samples, but will be governed by the same sampling distribution If we know the appropriate sampling distribution, we can reason about the underlying population based on the observed value of a statistic e.g. we calculate a sample mean from a random sample; in what range do we think the actual (population) mean really sits?

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9 M anufacturing Estimation and Confidence Intervals Point Estimation:
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lecture6 - MIT OpenCourseWare http/ocw.mit.edu 2.830J...

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