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Thepotemich [5.8K]
2 years ago
11

A confidence interval was used to estimate the proportion of statistics students that are female. A random sample of 72 statisti

cs students generated the following 90% confidence interval: (0.438, 0.642). Using the information above, what size sample would be necessary if we wanted to estimate the true proportion to within ±0.08 using 95% confidence? Explain/show how you obtain your answer
Mathematics
1 answer:
jeyben [28]2 years ago
3 0

Answer:

We need a sample of size at least 13.

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}}

90% confidence interval: (0.438, 0.642).

The proportion estimate is the halfway point of these two bounds. So

\pi = \frac{0.438 + 0.642}{2} = 0.54

95% confidence level

So \alpha = 0.05, z is the value of Z that has a pvalue of 1 - \frac{0.05}{2} = 0.975, so Z = 1.96.

Using the information above, what size sample would be necessary if we wanted to estimate the true proportion to within ±0.08 using 95% confidence?

We need a sample of size at least n.

n is found when M = 0.08. So

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

0.08 = 1.96\sqrt{\frac{0.54*0.46}{n}}

0.08\sqrt{n} = 1.96\sqrt{0.54*0.46}

\sqrt{n} = \frac{1.96\sqrt{0.54*0.46}}{0.08}

(\sqrt{n})^{2} = (\frac{1.96\sqrt{0.54*0.46}}{0.08})^{2}

n = 12.21

Rounding up

We need a sample of size at least 13.

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