Question

Some shrubs have the useful ability to resprout from their roots after their tops are destroyed. Fire is a particular threat to shrubs in dry climates, as it can injure the roots as well as destroy the aboveground material. One study of resprouting took place in a dry area of Mexico. The investigation clipped the tops of samples of several species of shrubs. In some cases, they also applied a propane torch to the stumps to simulate a fire. Of 18 specimens of a particular species, 5 resprouted after fire. Estimate with 99.5% confidence the proportion of all shrubs of this species that will resprout after fire.

Answer 1

The 99.5% confidence interval for the proportion of all shrubs of this species that will resprout after fire is (0, 0.5745).

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)`.

For this problem, we have that:

`n = 18, \pi = (5)/(18) = 0.2778`

99.5% confidence level

So
`\alpha = 0.005`, z is the value of Z that has a pvalue of
`1 - (0.005)/(2) = 0.9975`, so
`Z = 2.81`.

The lower limit of this interval is:

`\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.2778 - 2.81\sqrt{(0.2778*0.7222)/(18)} = -0.01 = 0`

We cannot have a negative proportion, so we use 0.

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.2778 + 2.81\sqrt{(0.2778*0.7222)/(18)} = 0.5745`

The 99.5% confidence interval for the proportion of all shrubs of this species that will resprout after fire is (0, 0.5745).