Question

. How large a sample is needed if we wish to estimate the proportion of American voters in favor of President Trump being reelected, we wish this estimate to be with in 0.038 of the true proportion, using a confidence level of 0.96. From the previous surveys, the proportion of American voters favoring his reelection to be 0.42?

Answer 1

We need a sample of size at least 712.

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{(\pi(1-\pi))/(n)}`

In which

z is the zscore that has a pvalue of
`1 - (\alpha)/(2)`.

The margin of error of an interval is:

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

96% confidence level:

So
`\alpha = 0.04`, z is the value of Z that has a pvalue of
`1 - (0.04)/(2) = 0.98`, so
`Z = 2.054`.

How large a sample is needed if we wish to estimate the proportion of American voters in favor of President Trump being reelected, we wish this estimate to be with in 0.038 of the true proportion

A sample of at least n, in which n is found when M = 0.038.

We have that
`\pi = 0.42`

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

`0.038 = 2.054\sqrt{(0.42*0.58)/(n)}`

`0.038√(n) = 2.054√(0.32*0.58)`

`√(n) = (2.054√(0.32*0.58))/(0.038)`

`n = 711.7`

Rounding up

We need a sample of size at least 712.