1 Name:___________________ID:__________________AMS______ Extra credit opportunity Enter your answers on the blanks provided below. Show any work on page 3: Infinite Finite a.___________ a.____________ c.___________ c.____________ d.___________ d.____________ e.___________ e.____________ f. f. Note: 2 =1.414, 3 / 2 =.866 Please give answers in terms of radicals. ______________________________________________________ The central limit theorem and the standard error of the mean σ (X) = σ (X) n are based on the premise that the population has infinitely many members. When samples are chosen with replacement the population is effectively infinite. Consider the population of N= 5 objects {1, 2, 3, 4, 5} 1. Find the population mean µ 2. Find the population std. deviation σ 3. Take (with replacement) samples of size n=2 from the above population. 2 (1,1) (1,2) (1,3) (1,4) (1,5) (2,1) (2,2)……….. _______________________________________________________ a. How many such samples are possible? b. List all the X ’s, i.e. the means of all these samples of size 2 _______________________________________________________ c. Find the mean of these means, i.e. E(X) d. Is E(X) = µ? e. Find the standard errorσ (X), i.e. the std. deviation of all these means σ (X) f. Show that the standard error equals 4. Many realistic applications involve sampling without replacement. For example, in manufacturing, quality control inspectors sample items from a finite production run without replacement. For such a finite population, we have to adjust the value of σ (X). Take (without replacement) samples of size 2 from the above population of N= 5 objects {1, 2, 3, 4, 5} (1,2) (1,3) (1,4) (1,5) (2,3), (2,4), (2,5)................ ______________________________________________________ a. How many such samples are possible? b. List all the X ’s, i.e. the means of all these samples of size 2 _______________________________________________________ c. Find the mean of these means, i.e. E(X) d. Is E(X) = µ? e. Find the standard errorσ (X), i.e. the std. deviation of all these means 3 f. Show that the standard error equals Note: N − n N −1 is called the finite population correction factor. Typically it is used when the sample size n is greater than 5% of the finite population size. 

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