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You are the operations manager for an airline and you are considering a higher fare level for passengers in aisle seats. How many randomly selected air passengers must you​ survey? Assume that a prior survey suggests that about 22​% of air passengers prefer an aisle seat. Assume that you want to be 98% confident that the sample percentage is within 5.1 percentage points of the true population percentage.

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Answer:

358 randomly selected air passengers must be surveyed.

Explanation:

In a sample with a number n of people surveyed with a probability of a success of
\pi, and a confidence level of
1-\alpha, we have the following confidence interval of proportions.


\pi \pm z\sqrt{(\pi(1-\pi))/(n)}

In which

z is the zscore that has a pvalue of
1 - (\alpha)/(2).

The margin of error is given by:


M = z\sqrt{(\pi(1-\pi))/(n)}

98% confidence level

So
\alpha = 0.02, z is the value of Z that has a pvalue of
1 - (0.02)/(2) = 0.99, so
Z = 2.327.

Assume that a prior survey suggests that about 22​% of air passengers prefer an aisle seat.

This means that
\pi = 0.22

How many randomly selected air passengers must you​ survey?

We need a sample of n, and n is found for which
M = 0.051. So


M = z\sqrt{(\pi(1-\pi))/(n)}


0.051 = 2.327\sqrt{(0.22*0.78)/(n)}


0.051√(n) = 2.327√(0.22*0.78)


√(n) = (2.327√(0.22*0.78))/(0.051)


(√(n))^2 = ((2.327√(0.22*0.78))/(0.051))^2


n = 357.2

Rounding up

358 randomly selected air passengers must be surveyed.

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