An experiment was designed to work as follows To start take a sample of 25 n

An experiment was designed to work as follows to

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An experiment was designed to work as follows: To start, take a sample of 25 = n observations from a population that follows the normal distribution with 5 = μ and 2 = σ . From the sample observations calculate the sample mean and a 90% confidence interval estimate. Now take another 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.
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Econ 325 – Chapter 7 17 Estimation results for 20 selected samples are given below. Sample x 90% Confidence Interval Estimate ) 25 2 ( 1.645 ± x 1 4.94 4.28 5.60 2 5.29 4.63 5.95 3 4.97 4.32 5.63 4 4.15 3.49 4.81 ** 5 4.49 3.84 5.15 6 5.06 4.40 5.72 7 5.35 4.69 6.00 8 5.28 4.62 5.93 9 4.83 4.17 5.48 10 5.49 4.83 6.15 11 4.71 4.06 5.37 12 4.71 4.05 5.37 13 5.59 4.94 6.25 14 4.92 4.26 5.58 15 4.75 4.09 5.40 16 5.54 4.89 6.20 17 5.04 4.39 5.70 18 4.72 4.06 5.38 19 4.95 4.29 5.60 20 5.84 5.18 6.50 ** Each interval estimate is centered at the calculated sample mean x . All interval estimates have the same width. The samples marked with ** have interval estimates that do not contain the true population mean 5 = μ . That is, Sample 4 has an upper limit below 5 = μ and Sample 20 has a lower limit that exceeds 5. It can be seen that the other 18 samples listed all have interval estimates that contain the true population mean 5. Econ 325 – Chapter 7 18 To demonstrate the interpretation of a 90% confidence interval, for the 1000 samples generated for the experiment, about 900 of the calculated interval estimates should contain the true population mean 5 and the remaining interval estimates (about 100) will not contain the true mean (like Sample numbers 4 and 20 in the list printed above). The computer experiment reported above counted 107 interval estimates that did not contain the population mean 5 = μ . It should be noted that if the experiment was repeated, a different set of 1000 samples would be generated, and therefore the numerical summary of the results would be a bit different.
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Econ 325 – Chapter 7 19 Now take another look at the calculation for the confidence interval estimate: n z x c σ ± The width of the interval estimate is: n z 2 c σ The width will be affected by: the level of α . This sets the value of c z . Smaller α leads to a wider confidence interval. That is, a 99% interval is wider than a 95% interval. the variance 2 σ . As 2 σ increases, the confidence interval becomes wider. the sample size n . As n increases, the confidence interval becomes narrower. In general, a wide confidence interval reflects imprecision in the knowledge about the population mean. Econ 325 – Chapter 7 20 Chapter 7.3 Interval Estimation Continued A 90% confidence interval estimate for the population mean can be calculated as: n x σ ± 1.645 In practice, the population variance 2 σ is unknown.
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