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chap07PRN econ 325

From a data set with n observations calculate the

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From a data set with n observations calculate the sample mean x and sample standard deviation s . A ) 1 ( α - 100 % confidence interval estimate for the population mean is given by: n s t x c ± where c t is the critical value from the t-distribution with ( n - 1 ) degrees of freedom such that: 2 ) t t ( P c ) 1 n ( α = > - Econ 325 – Chapter 7 26 Example: Gasoline Consumption of Trucks A data set has observations on fuel consumption, in miles per gallon, for 24 trucks. Summary statistics are: 18.68 = x and 1.695 = s A 90% confidence interval estimate for the population mean fuel consumption is: 24 1.695 18.68 c t ± The graph below illustrates the t-distribution critical value. PDF of ) 23 ( t tc 0 -tc Area = 0.9 Upper Tail Area = 0.05 Lower Tail Area = 0.05
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Econ 325 – Chapter 7 27 The Appendix Table for the t-distribution can be used to look-up the critical value c t . For this example, to correctly use the table, select the degrees of freedom ( n - 1 ) = 23, and set the upper tail area to 0.10/2 = 0.05. Alternatively, use Microsoft Excel. Select Insert Function: TINV(0.10, 23) This returns the answer: 1.714 = c t The calculations required for the interval estimate are: 24 1.695 1.714 18.68 ± For the given data set, the calculations give a 90% confidence interval estimate for the population mean as: [ 19.27 18.09 , ] Econ 325 – Chapter 7 28 Chapter 7.7 Sample Size Determination A wide confidence interval reflects uncertainty about the parameter being estimated. A larger sample size n will give a narrower interval. Consider a confidence interval for the population mean μ in a situation where the population variance 2 σ is known from previous research. For a given data set, a ) 1 ( α - 100 % interval estimate for the population mean is: [ n z x , n z x c c σ + σ - ] where c z is the value such that: 2 1 ) z ( F ) z Z ( P c c α - = = < The width of the interval estimate is: n z 2 w c σ = Suppose a set width w is desired. What sample size n will guarantee this width ? Rearranging gives: w z 2 n c σ = By squaring both sides: ( ) 2 c w z 2 n σ = Round up to get an integer number for n .
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