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dlinn [17]
3 years ago
12

What sample size is needed to give a margin of error within in estimating a population mean with 95% confidence, assuming a prev

ious sample had .
Mathematics
1 answer:
atroni [7]3 years ago
4 0

Answer:

n = (\frac{1.96\sqrt{\pi(1-\pi)}}{M})^2

The sample size needed is n(if a decimal number, round up to the next integer), considering the estimate of the proportion \pi(if no previous estimate use 0.5) and M is the desired margin of error.

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 z-score that has a p-value of 1 - \frac{\alpha}{2}.

The margin of error is of:

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

95% confidence level

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

Needed sample size:

The needed sample size is n. We have that:

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

M = 1.96\sqrt{\frac{\pi(1-\pi)}{n}}

\sqrt{n}M = 1.96\sqrt{\pi(1-\pi)}

\sqrt{n} = \frac{1.96\sqrt{\pi(1-\pi)}}{M}

(\sqrt{n})^2 = (\frac{1.96\sqrt{\pi(1-\pi)}}{M})^2

n = (\frac{1.96\sqrt{\pi(1-\pi)}}{M})^2

The sample size needed is n(if a decimal number, round up to the next integer), considering the estimate of the proportion \pi(if no previous estimate use 0.5) and M is the desired margin of error.

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Step-by-step explanation:

For this case we have the following function given:

y(t) = 140 (2.7)^{0.01t}

Where t represent the time in years

For this case this model follows the general model given by:

y = a b^t

Where a = 140 represent the initial amount of bisons and y the number of bisons after t years and b = 2.7 represent the growth rate

For this case we want to find the value of y when t =30, and if we replace the value of t =30 in our function we got:

y(30) = 140(2.7)^{0.01*30} = 188.598

And if we round to the nearest whole number we got 189 bisons after 30 years

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