6. It is election night and you are working for a TV news service in Florida. Yo

Question

6. It is election night and you are working for a TV news service in Florida. You are covering a race between two candidates, George and Al. Assume that all voters in the state of Florida vote for one of these two candidates and let p equal to the proportion of people who vote for Al. Votes are now being counted, and the network executives want you to make a projection for which candidate will win the state. Though of course they want this projection as soon as possible, they are concerned with the network's credibility and are willing to accept at most a 5% chance that the projection is wrong. (a) Suppose that 1000 votes from the state have been counted. The results are 526 votes for Al (the other 474 voted for George). Construct a 95% confidence interval for p. Can you project a winner in the election? (6pts) (b) Now suppose that 10,000 votes have been counted and 5,150 of them were votes for Al. Construct a 95% confidence interval for p. Can you project a winner in the election? Gipvies)

Explanation / Answer

Solution:

a) Here we have p = 526/1000 = 0.526, and n = 1000 So S.E.(p) = sqrt[0.526(10.526)/1000] = 0.01578.
The 95% CI is p ± 2 S.E.(p) = 0.526 ± 0.03156 = (0.49444, 0.55756)
This confidence interval contains the value p = 0.5. In other words, we can't tell based on this data whether more than 50% of all voters voted for Al. (We could also have done this by testing H0 : p = 0.5 directly; we'd fail to reject.)

b) This time p = 5150/10000 = 0.515, and n = 10, 000.
Now S.E.(p) = sqrt[0.515(10.515)/10,000] = 0.005
The 95% CI is p ± 2 S.E.(p) = 0.515 ± 0.005 = (0.505, 0.525) In particular, this CI gives us enough information (at the 95% level) to say that p is bigger than 0.5, or that more than 50% of people voted for Al.

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