Some shrubs have the useful ability to resprout from their roots after their top

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 12 specimens of a particular species, 7 resprouted after fire. Estimate with 99.9% confidence the proportion of all shrubs of this species that will resprout after fire. I NEED THE CONDITIONS FOR A LARGE SAMPLE. NOT JUST A NORMAL CONFIDENCE INTERVAL. thank you!

Explanation / Answer

Solution:

Even regarding the sample as an SRS from the population of that species, we cannot use the large-sample confidence inter-val. To do so requires that there be at least 12 'successes' (and 12 'failures'), but only 7 resprouted after fire.

The plus four confidence interval is still valid (the sample size is 10 or greater), and it is generally more reliable so we use it instead. Such a confidence interval looks like

= p ± z* .sqrt(p(1-p)/n+4)

where p is the proportion of successes to total elements when two imaginary successes and two imaginary failures are added. Here, p = 7+2/12+4 = 0.5625.

Using Table C or technology, we find the critical value e correponding to 99.9% confidence:

z* = 3.291. Thus, the plus four confidence interval is

p ± z* .sqrt(p(1-p)/n+4) = 0.5625 ± 3.291.sqrt(0.5625(1-0.5625)/16)

which, in interval notation, is [0.1544, 0.9706].

In other words, we are 99.9% confident that the true proportion of shrubs that resprout after fire is between [0.1544 and 0.9706.

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