# chap07 - Statistics for Business and Economics Chapter 7...

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Chap 7-1 Chapter 7 Estimation: Single Population Statistics for Business and Economics

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Chap 7-2 Chapter Goals After completing this chapter, you should be able to: Distinguish between a point estimate and a confidence interval estimate Construct and interpret a confidence interval estimate for a single population mean using both the Z and t distributions Form and interpret a confidence interval estimate for a single population proportion
Chap 7-3 Confidence Intervals Content of this chapter Confidence Intervals for the Population Mean, μ when Population Variance σ 2 is Known when Population Variance σ 2 is Unknown Confidence Intervals for the Population Proportion, (large samples) p ˆ

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Chap 7-4 Definitions An estimato r of a population parameter is a random variable that depends on sample information . . . whose value provides an approximation to this unknown parameter A specific value of that random variable is called an estimate
Chap 7-5 Point and Interval Estimates A point estimate is a single number, a confidence interval provides additional information about variability Point Estimate Lower Confidence Limit Upper Confidence Limit Width of confidence interval

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Chap 7-6 We can estimate a Population Parameter … Point Estimates with a Sample Statistic (a Point Estimate) Mean Proportion P x μ p ˆ
Chap 7-7 Unbiasedness A point estimator is said to be an unbiased estimator of the parameter θ if the expected value, or mean, of the sampling distribution of is θ , Examples: The sample mean is an unbiased estimator of μ The sample variance is an unbiased estimator of σ 2 The sample proportion is an unbiased estimator of P θ ˆ θ ˆ θ ) θ E( = ˆ

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Chap 7-8 is an unbiased estimator, is biased: 1 θ ˆ 2 θ ˆ θ ˆ θ 1 θ ˆ 2 θ ˆ Unbiasedness (continued)
Chap 7-9 Bias Let be an estimator of θ The bias in is defined as the difference between its mean and θ The bias of an unbiased estimator is 0 θ ˆ θ ˆ θ ) θ E( ) θ Bias( - = ˆ ˆ

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Chap 7-10 Consistency Let be an estimator of θ is a consistent estimator of θ if the difference between the expected value of and θ decreases as the sample size increases Consistency is desired when unbiased estimators cannot be obtained θ ˆ θ ˆ θ ˆ
Chap 7-11 Most Efficient Estimator Suppose there are several unbiased estimators of θ The most efficient estimator or the minimum variance unbiased estimator of θ is the unbiased estimator with the smallest variance Let and be two unbiased estimators of θ , based on the same number of sample observations. Then, is said to be more efficient than if The relative efficiency of with respect to is the ratio of their variances: ) θ Var( ) θ Var( 2 1 ˆ ˆ < ) θ Var( ) θ Var( Efficiency Relative 1 2 ˆ ˆ = 1 θ ˆ 2 θ ˆ 1 θ ˆ 2 θ ˆ 1 θ ˆ 2 θ ˆ

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chap07 - Statistics for Business and Economics Chapter 7...

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