Exam 3: Review of Statistics

The following types of statistical inference are used throughout econometrics,with the exception of

Assume that two presidential candidates,call them Bush and Gore,receive 50% of the votes in the population.You can model this situation as a Bernoulli trial,where Y is a random variable with success probability Pr(Y = 1)= p,and where Y = 1 if a person votes for Bush and Y = 0 otherwise.Furthermore,let be the fraction of successes (1s)in a sample,which is distributed N(p, )in reasonably large samples,say for n ≥ 40. (a)Given your knowledge about the population,find the probability that in a random sample of 40,Bush would receive a share of 40% or less. (b)How would this situation change with a random sample of 100? (c)Given your answers in (a)and (b),would you be comfortable to predict what the voting intentions for the entire population are if you did not know p but had polled 10,000 individuals at random and calculated ? Explain. (d)This result seems to hold whether you poll 10,000 people at random in the Netherlands or the United States,where the former has a population of less than 20 million people,while the United States is 15 times as populous.Why does the population size not come into play?

Let p be the success probability of a Bernoulli random variable Y,i.e. ,p = Pr(Y = 1).It can be shown that ,the fraction of successes in a sample,is asymptotically distributed N(p, .Using the estimator of the variance of , ,construct a 95% confidence interval for p.Show that the margin for sampling error simplifies to 1/ if you used 2 instead of 1.96 assuming,conservatively,that the standard error is at its maximum.Construct a table indicating the sample size needed to generate a margin of sampling error of 1%,2%,5% and 10%.What do you notice about the increase in sample size needed to halve the margin of error? (The margin of sampling error is 1.96×SE( ))