SOLUTION: 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?

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Question 1202810: 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 you want to be 95% confident that the sample percentage is within 3.5 percentage points of the true population percentage. Assume nothing is known about the percentage of passengers who prefer aisle seats. (The answer must be an integer)

i got this wrong the answer was 784 but i dont know how that is answers id like to know steps taken
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!

At 95% confidence, the critical z value is roughly z = 1.960. Use a table or stats calculator to determine this.

E = margin of error
E = 0.035
This is the decimal form of 3.5%

phat = sample proportion
phat's job is to estimate the population proportion (p).
Since we do not know the value of phat, we go with the most conservative estimate of phat = 0.5 which is right in the middle between 0 and 1.

Here are the values we'll plug in
z = 1.960
E = 0.035
phat = 0.5

Plug them into the formula below
n = phat*(1-phat)*(z/E)^2
n = 0.5*(1-0.5)*(1.96/0.035)^2
n = 784

In this case we landed exactly on 784 without needing to round.
In most cases we'll get some decimal value.
The idea is to round UP to the nearest integer. Always round up.
This is to clear the hurdle needed.

Answer: 784

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Extra info:

You may be wondering "where did that formula come from?"

It's the result of solving
E = z*sqrt(phat*(1-phat)/n)
for the variable n.
This second formula computes the margin of error for a confidence interval proportion.

Let's solve for n.
E = z*sqrt(phat*(1-phat)/n)
E/z = sqrt(phat*(1-phat)/n)
(E/z)^2 = phat*(1-phat)/n
(z/E)^2 = n/( phat*(1-phat) )
phat*(1-phat)*(z/E)^2 = n
n = phat*(1-phat)*(z/E)^2