// File MIT-Lab-03-04-00-02.txt. Edition 12/29/2012. // Lab Specs // Title Opinion_Poll_Simulation // List of Population Proportions .1 .9 .1 .5 // List of Sample Sizes 1_2_3_4_10_16_25_50_100_200_400 // List of From Bounds - Decimals 0.000 1.000 .025 -1.000 // List of To Bounds - Decimals 0.000 1.000 .025 -1.000 // List of From Bounds - Integers 0 -1 1 -1 // List of To Bounds - Integers 0 -1 1 -1 // Problem Specs // ActFrac // SampleSize // PauseCheckbox(-1 checked 0 cleared) // NonRandomSampleCheckbox(-1 visible unchecked, 0 not visible, 1 visible checked) // To-FromValues(0 hide, -1 reset values to none, 1 leave current values intact) ` .5 25 0 0 -1 Objective: Illustrate the Central Limit Theorem: Show that as the sample size increases the normal distribution becomes a better and better approximation of EstFrac's probability distribution. _ Note that .5 has been selected as the actual population fraction. That is, the election is a toss up: half the population supports Clint and half does not. ` .5 25 0 0 -1 1. Suppose that the sample size is 25. _ 1a. Using the appropriate equations, calculate the mean and variance of EstFrac's probability distribution. _ 1b. What is the standard deviation of EstFrac's probability distribution? ` Focus on the two lists in the lower left hand corner of the screen: the "From" list and the "To" list. When you specify a From and To value, the simulation calculates the percent of repetitions in which the numerical value of the estimated fraction falls between these two values. _ 1c. Based on your answers to 1a and 1b, what From and To values should you specify to calculate the percent of repetitions within 1 standard deviation of the mean? _ ` 1d. Specify these values. Click Start and after many repetitions, click Stop. What percent of repetitions fall within 1 standard deviations of the mean? _ 1e. Respecify the From and To values to determine the percent of repetitions that fall within 2 standard deviations of the mean. Click Start and then after many, many repetitions repetitions, click Stop. What is the percent? _ 1f. Respecify the From and To values to determine the percent of repetitions that fall within 3 standard deviations of the mean. Click Start and then after many, many repetitions repetitions, click Stop. What is the percent? ` .5 100 0 0 -1 2. Next, note that the sample size have been increased from 25 to 100. _ 2a. Using the appropriate equations, calculate the mean and variance of EstFrac's probability distribution. _ 2b. What is the standard deviation of EstFrac's probability distribution? _ 2c. Based on your above answers, what From and To values should you specify to calculate the percent of repetitions within 1 standard deviation of the mean? ` 2d. Specify these values. Click Start and after many repetitions, click Stop. What percent of repetitions fall within 1 standard deviations of the mean? _ 2e. Respecify the From and To values to determine the percent of repetitions that fall within 2 standard deviations of the mean. Click Start and then after many, many repetitions repetitions, click Stop. What is the percent? _ 2f. Respecify the From and To values to determine the percent of repetitions that fall within 3 standard deviations of the mean. Click Start and then after many, many repetitions repetitions, click Stop. What is the percent? ` .5 400 0 0 -1 3. Last, note that the sample size have been increased from 100 to 400. _ 3a. Using the appropriate equations, calculate the mean and variance of EstFrac's probability distribution. _ 3b. What is the standard deviation of EstFrac's probability distribution? _ 3c. Based on your above answers, what From and To values should you specify to calculate the percent of repetitions within 1 standard deviation of the mean? ` 3d. Specify these values. Click Start and after many repetitions, click Stop. What percent of repetitions fall within 1 standard deviations of the mean? _ 3e. Respecify the From and To values to determine the percent of repetitions that fall within 2 standard deviations of the mean. Click Start and then after many, many repetitions repetitions, click Stop. What is the percent? _ 3f. Respecify the From and To values to determine the percent of repetitions that fall within 3 standard deviations of the mean. Click Start and then after many, many repetitions repetitions, click Stop. What is the percent? ` .5 400 0 0 -1 4. In a single table, summarize your answers to 1d, 1e, 1f and 2d, 2e, 2f, and 3d, 3e, 3f. That is, for each sample size, 25, 100, and 400, what is the percent of repetitions that fall with _ 1 standard deviation of the mean _ 2 standard deviations of the mean _ 3 standard deviations of the mean after many, many repetitions. _ 5. What interesting observation does your table illustrate?