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gizmo_the_mogwai [7]
3 years ago
7

) Beth wants to determine a 99 percent confidence interval for the true proportion p of high school students in the area who att

end their home basketball games. Out of n randomly selected students she finds that that exactly half attend their home basketball games. About how large would n have to be to get a margin of error less than 0.01 for p
Mathematics
1 answer:
Lunna [17]3 years ago
6 0

Answer:

n has to be at least 16577.

Step-by-step 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{\frac{\pi(1-\pi)}{n}}

In which

z is the zscore that has a pvalue of 1 - \frac{\alpha}{2}.

The margin of error is:

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

Out of n randomly selected students she finds that that exactly half attend their home basketball games.

This means that \pi = 0.5

99% confidence level

So \alpha = 0.01, z is the value of Z that has a pvalue of 1 - \frac{0.01}{2} = 0.995, so Z = 2.575.

About how large would n have to be to get a margin of error less than 0.01 for p

We have to find n for which M = 0.01. So

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

0.01 = 2.575\sqrt{\frac{0.5*0.5}{n}}

0.01\sqrt{n} = 2.575*0.5

\sqrt{n} = \frac{2.575*0.5}{0.01}

(\sqrt{n})^2 = (\frac{2.575*0.5}{0.01})^2

n = 16576.6

Rounding up

n has to be at least 16577.

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