Description

1. Go to http://www.pollingreport.com and find a polling statistic that interests you most. Be sure to click on the “full details” option, where available, to get the sample size for the survey item. Then calcualte the 95 percent and 99 percent confidence intervals for the population value of the statistic you have in mind, showing all of your work. Take a screen shot of the page from the web site and paste it into your knitted Word file when you submit.

2. For the same survey item, what would happen to the confidence interval if the sample size were cut in half? What would happen instead if it were doubled? Assume that the sample standard deviation does not change and show your work.

3. A Probability Model for Betting Market Election Prediction

Earlier in this chapter, we used preelection polls with a probability model to predict Obama’s electoral vote share in the 2008 US election. In this exercise, we will apply a similar procedure to the Intrade betting market data analyzed in an exercise in chapter 4 (see section 4.5.1). The 2008 Intrade data set are available in table 4.9. Recall that each row of the data set represents daily trading information about the contracts for either the Democratic or Republican nominee’s victory in a particular state. The 2008 election results data are available as pres08.csv, with variable names and descriptions appearing in table 4.1.

1.          We analyze the contract of the Democratic Party nominee winning a given state . Recall from section 4.5.1 that the data set contains the contract price of the market for each state on each day  leading up to the election. We will interpret PriceD as the probability  that the Democrat would win state  if the election were held on day . To treat PriceD as a probability, divide it by 100 so it ranges from 0 to 1. How accurate is this probability? Using only the data from the day before the election (November 4, 2008) within each state, compute the expected number of electoral votes Obama is predicted to win and compare it with the actual number of electoral votes Obama won. Briefly interpret the results. Recall that the actual number of electoral votes for Obama is 365, not 364, which is the sum of electoral votes for Obama based on the results data. The total of 365 includes a single electoral vote that Obama garnered from Nebraska’s 2nd Congressional District. McCain won Nebraska’s 4 other electoral votes because he won the state overall.

data(pres08, package = "qss")

2.          Next, using the same set of probabilities used in the previous question, simulate the total number of electoral votes Obama is predicted to win. Assume that the election in each state is a Bernoulli trial where the probability of success (Obama winning) is . Display the results using a histogram. Add the actual number of electoral votes Obama won as a solid line. Briefly interpret the result.

3.          In prediction markets, people tend to exaggerate the likelihood that the training or “long shot” candidate will win. This means that candidates with low (high)  have a true probability that is lower (higher) than their predicted . Such a discrepancy could introduce bias into our predictions, so we want to adjust our probabilities to account for it. We do so by reducing the probability for candidates who have a less than 0.5 chance. We will calculate a new probability  where  is the CDF of a standard normal random variable and  is its inverse, the quantile function. The R functions pnorm() and qnorm() can be used to compute  and , respectively. Plot , used in the previous questions, against . In addition, plot this function itself as a line. Explain the nature of the transformation.

4.          Using the new probabilities , repeat questions 1 and 2. Do the new probabilities improve predictive performance?

5.          Compute the expected number of Obama’s electoral votes using the new probabilities  for each of the last 120 days of the campaign. Display the results as a time-series pot. Briefly interpret the plot.

6.          For each of the last 120 days of the campaign, conduct a simulation as in question 2, using the new probabilities . Compute the quantile of Obama’s electoral votes at 2.5% and 97.5% for each day. Represent the range from 2.5% to 9.75 for each day as a vertical line, using a loop. Also, add the estimated total number of Obama’s electoral votes across simulations. Briefly interpret the result.