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. ______________________________________________________ (X) are n based on the premise that the population has infinitely many members. When samples are chosen with replacement the population is effectively infinite. The central limit theorem and the standard error of the mean (X) = Consider the population of N= 5 objects {1, 2, 3, 4, 5} 1. Find the population mean 2. Find the population std. deviation 3. Take (with replacement) samples of size n=2 from the above population. 1 (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) = ? e. Find the standard error (X) , i.e. the std. deviation of all these means (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 (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) = ? e. Find the standard error (X) , i.e. the std. deviation of all these means 2 f. Show that the standard error equals N n is called the finite population correction factor. Typically it is N 1 used when the sample size n is greater than 5% of the finite population size. Note: 3