2 2 thesamplesizenasnincreasestheconfidenceinterval

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 s= ∑ (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...
View Full Document

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.

Ask a homework question - tutors are online