Question

In 2006, the Florida Poll conducted by Florida International University asked whether current environmental regulations are too strict or not too strict. Of 1200 respondents, 229 said they were too strict. Find and interpret a (a) 95%, (b) 99% confidence interval for a relevant parameter at the time of that survey.

Answer 1

a) The 95% confidence interval for the true proportion of respondents who think current environmental regulations are too strict or not too strict is (0.1678, 0.2122). This means that we are 95% sure that the true proportion of respondents who think current environmental regulations are too strict or not too strict is between 0.1678 and 0.2122.

b) The 99% confidence interval for the true proportion of respondents who think current environmental regulations are too strict or not too strict is (0.1608, 0.2192). This means that we are 99% sure that the true proportion of respondents who think current environmental regulations are too strict or not too strict is between 0.1608 and 0.2192.

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)`.

For this problem, we have that:

`n = 1200, \pi = (229)/(1200) = 0.19`

(a) 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`.

The lower limit of this interval is:

`\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.19 - 1.96\sqrt{(0.19*0.81)/(1200)} = 0.1678`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.19 + 1.96\sqrt{(0.19*0.81)/(1200)} = 0.2122`

The 95% confidence interval for the true proportion of respondents who think current environmental regulations are too strict or not too strict is (0.1678, 0.2122). This means that we are 95% sure that the true proportion of respondents who think current environmental regulations are too strict or not too strict is between 0.1678 and 0.2122.

(b) 99% confidence interval.

So
`\alpha = 0.01`, z is the value of Z that has a pvalue of
`1 - (0.01)/(2) = 0.995`, so
`Z = 2.575`.

The lower limit of this interval is:

`\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.19 - 2.575\sqrt{(0.19*0.81)/(1200)} = 0.1608`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.19 + 2.575\sqrt{(0.19*0.81)/(1200)} = 0.2192`

The 99% confidence interval for the true proportion of respondents who think current environmental regulations are too strict or not too strict is (0.1608, 0.2192). This means that we are 99% sure that the true proportion of respondents who think current environmental regulations are too strict or not too strict is between 0.1608 and 0.2192.