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Statistics and Data Analysis. Professor William Greene Stern School of Business IOMS Department Department of Economics. The Margin of Error. The CNN/Opinion Research Corp. said 51 percent of those polled thought Biden did the best job, while 36 percent thought Palin did the best job.
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Statistics and Data Analysis Professor William Greene Stern School of Business IOMS Department Department of Economics
The Margin of Error The CNN/Opinion Research Corp. said 51 percent of those polled thought Biden did the best job, while 36 percent thought Palin did the best job. On the question of the candidates' qualifications to assume the presidency, 87 percent of those polled said Biden is qualified and 42 percent said Palin is qualified. The poll had a margin of error of plus or minus 4 percentage points. http://www.cnn.com/2008/POLITICS/10/03/debate.poll/index.html (9:30 AM, Friday, October 3, 2008)
What Does the “Margin of Error” Tell You? • Did Biden do the better job? If we could ask every individual all over the world who had an opinion, the proportion who think yes would be θ. • We assume that such a value exists. • It has to be as of a moment in time. The next day, the same question might get a different answer.
The Margin of Error • We can’t ask everyone, so we ask a sample of people, i = 1,…,n. • Do you think Biden did the better job? Xi = 1 if the person answers yes, Xi = 0 if they answer no. • 51% said yes, so P =(1/n)Σi xi = 0.51 • Is π = 0.51? No, we didn’t ask everyone, we just asked a sample. 0.51 is an estimate of π.
Why the 4% Margin of Error? • We acknowledge that the 0.51 might be inaccurate because it is based on a sample. • We assume that whatever n is, the sample was drawn randomly. • We use our empirical rule to figure out what the real value of π might be.
Some Theory • P = (1/n)Σi xi = 0.51 • P is a random variable. It is the sum of n Bernoulli random variables, divided by n. • E[xi] = the probability that person i will answer yes. This is π. So, the expected value of P is (1/n)Σi θ = θ. • They think P is a good estimate of θ.
Theory Continued • Since P is a random variable it has a variance. • Var[P] = (1/n2) Σi θ(1- θ) = θ(1- θ)/n • The standard deviation is the square root. • Use P to estimate this. The estimated standard deviation is sqr(.51(.49)/n).
That Margin of Error • They use the same empirical rule we do. • The margin of error is ±.04 is ±2 standard deviations. So, one standard deviation is .02. • .022 = P(1-P)/n. If P is .51, n = 625. • (According to David Gregory, they asked 560).
What they Found • Based on a survey of 625 people, we believe the proportion of people who think Biden did a better job is between 47% and 55%. • Is this CERTAIN? • Based on the same logic, the proportions for Palin are 28% to 36%. • Is this CERTAIN? • What would these ranges be if they had asked 6,250 people instead of 625?
