SOLUTION: A survey was given to a random sample of 105 voters in the United States to ask about their preference for a presidential candidate. Of those surveyed, 84 respondents said that the

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Question 1210248: A survey was given to a random sample of 105 voters in the United States to ask about their preference for a presidential candidate. Of those surveyed, 84 respondents said that they preferred Candidate A. At the 95% confidence level, what is the margin of error for this survey expressed as a proportion to the nearest thousandth?
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!

n = 105 = sample size
x = 84 = number of people who prefer candidate A
phat = x/n = 84/105 = 0.8 = sample proportion of those who prefer candidate A
The job of phat is to estimate the population proportion p.
The name "phat" is due to a hat over top the letter p.

At 95% confidence, the z critical value is approximately z = 1.96
This value is used very frequently in statistics that I recommend you memorize it.
Alternatively you can have it on a reference sheet or look it up online.

So we have these key values:
z = 1.96 (approx)
phat = 0.8
n = 105

Those values lead to:
E = margin of error for the sample proportion
E = z*sqrt(phat*(1-phat)/n)
E = 1.96*sqrt(0.8*(1-0.8)/105)
E = 0.076510565719 approximately
E = 0.077 when rounding to the nearest thousandth, i.e. 3 decimal places.

Answer: 0.077