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vovangra [49]
3 years ago
13

Exercise 6.13 presents the results of a poll evaluating support for the health care public option in 2009, reporting that 52% of

Independents in the sample opposed the public option. If we wanted to estimate this number to within 1% with 90% confidence, what would be an appropriate sample size?
Mathematics
1 answer:
sleet_krkn [62]3 years ago
3 0

Answer:

A sample size of 6755 or higher would be appropriate.

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 M is given by:

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

90% confidence level

So \alpha = 0.1, z is the value of Z that has a pvalue of 1 - \frac{0.1}{2} = 0.95, so Z = 1.645.

52% of Independents in the sample opposed the public option.

This means that p = 0.52

If we wanted to estimate this number to within 1% with 90% confidence, what would be an appropriate sample size?

Sample size of size n or higher when M = 0.01. So

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

0.01 = 1.645\sqrt{\frac{0.52*0.48}{n}}

0.01\sqrt{n} = 0.8218

\sqrt{n} = \frac{0.8218}{0.01}

\sqrt{n} = 82.18

\sqrt{n}^{2} = (82.18)^{2}

n = 6754.2

A sample size of 6755 or higher would be appropriate.

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