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 19 specimens of a particular species, 6 resprouted after fire. Estimate with 96% confidence the proportion of all shrubs of this species that will resprout after fire. Interval: .1884 to

Answer 1

Answer:

Step-by-step explanation:

Confidence interval is written as

Sample proportion ± margin of error

Margin of error = z × √pq/n

Where

z represents the z score corresponding to the confidence level

p = sample proportion. It also means probability of success

q = probability of failure

q = 1 - p

p = x/n

Where

n represents the number of samples

x represents the number of success

From the information given,

n = 19

x = 6

p = 6/19 = 0.32

q = 1 - 0.32 = 0.68

To determine the z score, we subtract the confidence level from 100% to get α

α = 1 - 0.96 = 0.04

α/2 = 0.04/2 = 0.02

This is the area in each tail. Since we want the area in the middle, it becomes

1 - 0.02 = 0.98

The z score corresponding to the area on the z table is 2.05. Thus, Thus, the z score for a confidence level of 96% is 2.05

Therefore, the 96% confidence interval is

0.32 ± 2.05√(0.68)(0.32)/19

The lower limit of the confidence interval is

0.32 - 0.22 = 0.1

The upper limit of the confidence interval is

0.32 + 0.22 = 0.54

Therefore, with 96% confidence interval, the proportion of all shrubs of this species that will resprout after fire is between 0.1 and 0.54