chap07PRN econ 325

# That is sample 4 has an upper limit below 5 μ and

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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. With a sample of data, a way to proceed is to calculate a variance estimate as: = - - = n 1 i 2 i 2 ) x x ( 1 n 1 s 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: n s x 1.645 ± 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. The use of a sample variance means that the critical value 1.645 may be smaller than what will give a correct 90% confidence interval.
Econ 325 – Chapter 7 21 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 interval is given with: n s x 1.645 ± Many economic data sets meet this requirement of a sample size exceeding 60 observations. However, methods are available for the calculation of exact interval estimates. These methods are standard features of computer software designed for the statistical analysis of economic data.

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That is Sample 4 has an upper limit below 5 μ and Sample...

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