On February 8, 2002, the Gallup Organization released the results of a poll conc

Question

On February 8, 2002, the Gallup Organization released the results of a poll concerning American attitudes toward the 19th Winter Olympic Games in Salt Lake City, Utah. The poll results were based on telephone interviews with a randomly selected national sample of 991 adults, 18 years and older, conducted February 4-6, 2002. Suppose we wish to use the poll's results to justify the claim that more than 27 percent of Americans (18 years or older) say that figure skating is their favorite Winter Olympic event. The poll actually found that 28 percent of a sample of 991 respondents reported that figure skating was their favorite event. If, for the sake of argument, we assume that 27 percent of Americans (18 years or older) say figure skating is their favorite event (that is, p = .27), calculate the probability of observing a sample proportion of .28 or more; that is, calculate P(Picture > .28). (Round standard deviation and probability answers to 4 decimal places. Round z-values to 2 decimal places.)

Explanation / Answer

This is a binomial problem. Sample size, n = 991. Probability of success, p = 0.27.

The question is asking what the probability is that the proportion of people in this sample that like figure skating is 0.28 or more.

Because of the large sample size, you can use a normal approximation. Sample mean of the proportion is just the given proportion of the population, 0.27.

Standard deviation is sqrt(p(1-p)/n) = sqrt(0.27*0.73/991) = 0.0141. (Remember that you are using the formulas for proportion, not counts.)

So now you have a normal distribution N(0.27,0.0141).

To find the probability of observing 0.28 or more,

find a z-score:
z = (x - mean)/std deviation = (0.28 - 0.27)/0.0141 = 0.7092 =0.71.

Use a z-table to find the associated p value, which is in this case 0.7611.

That is your left-hand proportion, and since you want the probability of getting 0.28 or MORE, it is 1 - 0.7611 = 0.2389.

Therefore probability (p0.28)=0.2389.

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