Question

We are studying the proportion of cities that have private garbage collector. We want a maximum error of 0.10 and a 95% confidence level, and tentatively estimate the proportion at 0.50. What is the minimum number of cities we need to contact?

Answer 1

The minimum number of cities we need to contact is 96.

Explanation:

In a sample with a number n of people surveyed with a probability of a success of
`\pi`, and a confidence interval
`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 is:

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

95% confidence interval

So
`\alpha = 0.05`, z is the value of Z that has a pvalue of
`1 - (0.05)/(2) = 0.975`, so
`Z = 1.96`.

In this problem, we have that:

`p = 0.5, M = 0.1`

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

`0.1 = 1.96*\sqrt{(0.5*0.5)/(n)}`

`0.1√(n) = 0.98`

`√(n) = 9.8`

`(√(n))^(2) = (9.8)^(2)`

`n = 96`

The minimum number of cities we need to contact is 96.