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Estimation of Population Mean: Sampling Distribution, Confidence Intervals, and Sample Size Determination

A solved assignment on population mean estimation and confidence intervals.

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Estimation of Population Mean: Sampling Distribution, Confidence Intervals,and Sample Size Determination1. (Sampling distribution & confidence interval) A random variable has a population mean equalto 1,573 and population variance equal to952,021. Your interest lies in estimating the populationmean of this random variable. (Of course, you do not know what the population mean andpopulation variance are.) With that in mind, you take a representative sample of size 85 from thepopulation of the random variable. You then use this sample data to calculate the sample averageas an estimate for the population mean.1.(a)Using your knowledge about the central limit theorem (CLT), and assuming that theCLT has already β€œestablished itself” / β€œkicked in” when the sample size is 85, what isthe probability that the sample average that you calculated will lie between 1,502 and1,748?In this case,𝑋ഀ~π‘π‘œπ‘Ÿπ‘šπ‘Žπ‘™α‰†1573,ࢧଽ5202185ቇRequired probability=𝑃(1502<𝑋ഀ<1748)=𝑃(𝑋ഀ<1748)βˆ’π‘ƒ(𝑋ഀ<1502)=π‘ƒα‰Œπ‘<1748βˆ’1573ΰΆ§ΰ°΅5202185α‰βˆ’π‘ƒα‰Œπ‘<1502βˆ’1573ΰΆ§ΰ°΅5202185ቍ= P(Z<1.65)–P(Z<-0.67)= 0.9505-0.2514= 0.69912.(b)You then use your sample data to calculate a 92% confidence interval for thepopulation mean. Assuming that you can estimate the variance of the underlyingrandom variable very precisely with your sample data, what should be the width of thisconfidence interval?Thewidth of this confidence intervalwould be twice the size of the margin of errorwhich is = 2*𝑍0.04βˆ—ΰΆ§ΰ―(𝑋)ΰ―‘=2βˆ—1.75βˆ—ΰΆ§ΰ¬½5202185=370.4093

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Statistics

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