Chapter 7

Chapter 7 - Chapter 7 Sampling Distributions and Confidence...

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Chapter 7: Sampling Distributions and Confidence Intervals Control Chart – looks at variation in data from samples of products taken over time. _ Sample Mean – X Motivation for Point Estimators Point Estimate – Single number calculated from sample data. It is used to estimate a parameter of the population. Point Estimator – formula or rule that is used to calculate the point estimate for a particular set of data. Sample statistics become point estimates. Common Point Estimators Two major categories used to classify variables: Qualitative Quantitative Point Estimators for Quantitative Variables If you are studying a single quantitative variable you typically wish to know value of population mean and value of population standard deviation. (Center and Variability) Wish to know values of those parameters that describe the behavior of the population. Population Mean – μ Population Standard Deviation – σ Use subtraction to compare two population means. Use ratio to compare variability o To compare the variation in two populations you must compare the variances. o Variance is just the standard deviation squared. Point Estimators for Qualitative Variables Wish to know what proportion or percentage of population has specific characteristic. Population percentage is labeled π. Use sample proportion, p, to estimate π. Typically Unknown Use point estimates to estimate values Use sample mean X to estimate μ. Use sample standard deviation to estimate σ.

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Summary of Commonly Used Point Estimators Type of Variable(s) Population Parameter (unknown, what you wish to estimate) Point Estimator or Sample Statistic (calculated from sample data) A single quantitative variable Population mean, μ _ Sample mean, X Two quantitative variables Difference in population means, μ 1 – μ 2 Ratio of population variances, σ 2 1 / σ 2 2 Differenc e in sample means, X 1 – X 2 Ratio of sample variances, s 2 1 /s 2 2 A single qualitative variable Population proportion, π Sample proportion, p Two qualitative variables Difference in population proportions, π 1 – π 2 Difference in sample proportions, p 1 – p 2 The mean the median and the mode all fit the definition of a point estimator. Point Estimator Qualities Should fairly estimate the unknown population parameter. Should yield a number close to the unknown population parameter as the sample size increases. The point estimator should not have a great deal of variability. More precisely stated: The point estimator should be unbiased. The point estimator should be consistent. The point estimator should be efficient. Unbiased Estimator – yields an estimate that is fair. Neither systematically overestimates the parameter nor systematically underestimates the parameter.
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