15 chapter 8

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Unformatted text preview: A 90% confidence interval estimate is calculated as: x ± 1.645 σ n What is the interpretation of an interval estimate ? In applied work, the calculated interval estimate is based on one sample of data. It may contain the true parameter μ , or it may not contain μ . Since the true value of μ is unknown, it is impossible to say whether or not the population mean is contained in the interval estimate calculated from the sample of data. 14 Chapter 8 The interpretation of a 90 % confidence interval estimate for the population mean can be explained in the context of repeated sampling. Different samples of data will give different calculated sample means. Some of the sample means will be less than the true mean μ and some will be greater than this value. Therefore, different samples will give different interval estimates. In a ‘large’ number of samples 90% of the interval estimates will contain the true population mean μ and the other 10% will not contain this value (that is, they either have an upper limit below μ or a lower limit that exceeds μ ). 15 Chapter 8 The interpretation of a 90 % confidence interval estimate can be demonstrated with a computer simulation. An experiment was designed to work as follows: To start, take a sample of n = 25 observations from a population that follows the normal distribution with μ = 5 and σ = 2 . From the sample observations calculate the sample mean and a 90% confidence interval estimate. Now takeanother sample of 25 observations and repeat the calculations. Continue drawing samples from the population. Stop after interval estimates from 1000 samples have been calculated. The calculation of the sample mean is based on an unbiased estimation rule. This says that the average of the 1000 calculated sample means should give the true mean 5. A computer experiment was tried and the results showed the average of the sample means was 5.003 to support the idea of an unbiased estimation rule. 16 Chapter 8 Estimation results for 20 selected samples are given below. Sample 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 x 90% Confidence Interval Estimate x ± 1.645 ( 2 4.94 5.29 4.97 4.15 4.49 5.06 5.35 5.28 4.83 5.49 4.71 4.71 5.59 4.92 4.75 5.54 5.04 4.72 4.95 5.84 4.28 4.63 4.32 3.49 3.84 4.40 4.69 4.62 4.17 4.83 4.06 4.05 4.94 4.26 4.09 4.89 4.39 4.06 4.29 5.18 25 ) 5.60 5.95 5.63 4.81 ** 5.15 5.72 6.00 5.93 5.48 6.15 5.37 5.37 6.25 5.58 5.40 6.20 5.70 5.38 5.60 6.50 ** Each interval estimate is centered at the calculated sample mean x . All interval estimates have the same widt...
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