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Unformatted text preview: h. The samples marked with ** have interval estimates that do not contain the true population mean μ = 5 . That is, Sample 4 has an upper limit below μ = 5 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. 17 Chapter 8 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 μ = 5 . 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. 18 Chapter 8 Now take another look at the calculation for the confidence interval estimate: x ± zc σ n The width of the interval estimate is: 2 ⋅ zc σ n The width will be affected by: • the level of α . This sets the value of zc . Smaller α leads to a wider confidence interval. That is, a 99% interval is wider than a 95% interval. • the variance σ . As σ increases, the confidence interval becomes wider. 2 2 • 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. 19 Chapter 8 Chapter 8.3 Interval Estimation Continued A 90% confidence interval estimate for the population mean can be calculated as: x ± 1.645 σ n In practice, the population variance σ is unknown. 2 With a sample of data, a way to proceed is to calculate a variance estimate as: 1n
∑ (x i − x )2 n − 1 i=1
2 Then, for the calculation of an interval estimate, replace the unknown σ with the calculated standard deviation s to get the interval estimate for the population mean as: x ± 1.645 s n A problem with this is that the confidence level is no longer guaranteed to be 0.90. The interval estimate may now be viewed as an approximate 90% interval estimate. Since s is an estimate of the population variance, the critical value 1.645 may be smaller than what will give a correct 90% confidence interval. 20 Chapter 8 It turns out that the quality of the approximation gradually improves with increasing sample size n. As a rough guideline, with n > 60, a good approximation for a 90% confidence interva...
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This note was uploaded on 02/06/2014 for the course ECON ECON 325 taught by Professor Whistler during the Spring '10 term at UBC.
- Spring '10