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05C_NoEx_F10 - Sample Proportion b p = X n where X is...

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The Hong Kong University of Science & Technology ISMT 111: Business Statistics Tutorial Set 5 Sampling Distribution of Sample Statistics The sampling distribution of a statistic is the probability distribution of all the possible values of this statistic which are computed from random samples with same sample size ( n ) . Notation: n = Sample size X = Random variable µ = Population mean X = Sample mean σ = Population standard deviation s = Sample standard deviation se = Standard error p = Population proportion b p = Sample proportion Central Limit Theorem (CLT) Regardless of the population distribution of X which with a mean µ and a standard deviation σ , if the sample size n is su ciently large ( n > 30 ), the sampling distribution of the sample mean X will be APPROXIMATELY normally distributed with the mean µ X = µ , and the standard deviation σ X = σ n when n > 30 X CLT N ( µ, σ 2 n ) Note : There are four possible cases: Normal Population Nonnormal Population n is large ( n > 30) X N ( µ, σ 2 n ) X CLT N ( µ, σ 2 n ) n is small ( n < 30) X N ( µ, σ 2 n ) Not Capable to Handle 1
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Sampling Distribution of a Sample Proportion
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Unformatted text preview: Sample Proportion: b p = X n where X is number of success in the sample • When n > 30 , by CLT b p ∼ N ( p, pq n ) Estimation • Estimator - a sample statistic that is used to estimate an unknown population parameter. • Estimate - an actual numerical value obtained for an estimator. • For example, if X = 10 , X is an estimator; ‘10’ is an estimate. Point Estimator Criteria of a Good Estimator • No Bias — E ( estimator ) = population parameter — For example, E ¡ X ¢ = µ , so X is an unbiased estimator for µ. • Small Variance — The estimators vary about the true parameter & this variation should be small i.e. standard error of the estimators should be small — V ar ( X ) < V ar ( e X ) where e X is any other estimator of population mean — Therefore, an unbiased estimator with smaller variance is closer to the population parameter than the one with larger variance. 2...
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05C_NoEx_F10 - Sample Proportion b p = X n where X is...

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