4.55 Normal approximation for a sample proportion. A sample survey contacted an SRS of 700 registered voters in Oregon shortly after an election and asked respondents whether they had voted. Voter records show that 56% of registered voters had actually voted. We will see in the next chapter that, in this situation, the proportion p^ of the sample who voted has approximately the Normal distribution with mean

μ=0.56 and standard deviation σ=0.019 .

a. If the respondents answer truthfully, what is P(0.52≤p^≤0.60) ?

This is the probability that the p^ estimates 0.56 within plus or minus 0.04.

b. In fact, 72% of the respondents said they had voted

(p^=0.72) . If respondents answer truthfully, what is P(p^≥0.72) ? This probability is so small that it is good evidence that some people who did not vote claimed that they did vote.