// File MIT-Lab-02-04-00-02.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 // PopFrac // SampleSize // PauseCheckbox(-1 checked 0 cleared) // NonRandomSampleCheckbox(-1 visible unchecked, 0 not visible, 1 visible checked) // To-FromValues(-1 reset to none, 0 leave current values intact) ` .5 1 -1 0 -1 Objective: Illustrate the relative frequency interpretation of probability. After many, many repetitions of the experiment the distribution of a random variable's numerical values mirrors the random variable's probability distribution; consequently, after many, many repetitions the mean and variance of the numerical values will equal the mean and variance of the probability distribution. ` .5 1 -1 0 -1 We begin by considering a sample size of only 1. Let v equal the number of individuals polled who support Clint. That is, v equals 1 if the individual polled supports Clint and 0 if the individual does not. _ As before, suppose that the election is a toss up; that is, suppose that the population fraction, ActFrac, equals .5. Half the population supports Clint and half does not. ` .5 1 -1 0 -1 1. What is the probability that v would equal each of its possible values, 0 and 1, in a single poll. What is the mean and variance of the random variable v's probability distribution? _ 2. Click the Start button. Does the individual support Clint? Record the numerical value of v for the first repetition of the experiment. ` .5 1 -1 0 -1 3. Click the Continue button to repeat the poll for a second time. What does the numerical value of v equal for the second repetition of the experiment? Record its value. Is the simulation computing the relative frequencies, the mean, and the variance of v's numerical values correctly? _ 4. Click the Continue button a few more times and record the value of v in each repetition. Convince yourself that the simulation is computing the relative frequencies, the mean, and the variance of v's numerical values correctly. ` .5 1 0 0 -1 5. Note that the Pause checkbox is cleared. Click the Continue button and then after many, many repetitions click the Stop button. What are the relative frequencies of v's numerical values? What are the mean and variance of the numerical values? _ 6. Compare your answers to 1 and 5. Are your answers consistent with the relative frequency interpretation of probability? Explain.