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W4INSE6220

# W4INSE6220 - 1 3 Moments of the population vs sample...

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1 INSE 6220 -- Week 4 Advanced Statistical Approaches to Quality Inferences about Process Control Sampling and Estimation Confidence intervals Control Charts and hypothesis testing Statistical basis for Control Charts Dr. A. Ben Hamza Concordia University 2 Using the normal cdf and pdf We often want to talk about “percentage points” of the distribution-portion in the tails. >> Also, we have: Example: / 2 / 2 / 2 / 2 -1 / 2 ( ) 1 ( ) 1 ( ) 2 ( ) 1 2 = 1 2 P Z z P Z z z z z     icdf('normal',1- /2,0,1) /2 / 2 ( ) ( ) 2 P Z z z      0.20/ 2 0.10 0.05/ 2 0.025 1.2816 1.96 z z z z 3 Moments of the population vs. sample statistics 1 2 2 2 2 2 2 1 2 2 2 1 ( ) 1 ( ) ( ) 1 ( ) ( ) n X i i n X X X i i E X X X n Var X E X S S X X n E X E X 2 2 2 1 2 2 1 ( , ) ( )( ) 1 ( ) ( ) ( ) ( , ) ( ) ( ) n XY X y XY i i i XY XY XY XY X Y X Y S S Cov X Y E X Y S X X Y Y n E XY E X E Y S Cov X Y r S S Var X Var Y   Mean Variance Standard Deviation Covariance Correlation Coefficient Population Sample 4 Statistical Inference The purpose of statistical inference is to obtain information about a population from information contained in a sample. A population is the set of all the elements of interest. A sample is a subset of the population. The sample results provide only estimates of the values of the population characteristics. A parameter is a numerical characteristic of a population. With proper sampling methods , the sample results will provide “good” estimates of the population characteristics. In point estimation we use the data from the sample to compute a value of a sample statistic that serves as an estimate of a population parameter. We refer to as the point estimator of the population mean . s is the point estimator of the population standard deviation . When the expected value of a point estimator is equal to the population parameter, the point estimator is said to be unbiased . X

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5 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 A statistic is a random variable, which itself has a sampling 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 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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