Question

Question The average number of acres burned by forest and range fires in a county is 4,500 acres per year, with a standard deviation of 780 acres. The distribution of the number of acres burned is normal. What is the probability that between 3,000 and 4,800 acres will be burned in any given year? Round your answer to four decimal places. Provide your answer below:​

Answer 1

The probability that between 3,000 and 4,800 acres will be burned in any given year is 0.6206

Explanation:

Given

Mean = μ = 4500 acres

SD = σ = 780 Acres

Two data points between which the probability has to be calculated are:

3000 and 4800

x1 = 3000

x2 = 4800

z-score will be found for both data points

z-score is given by the formula

`z = (x-mean)/(SD)`

The z-scores are:

`z_1 = (x_1-mean)/(SD) = (3000-4500)/(780) = (-1500)/(780) = -1.92\\z_2 = (x_2-mean)/(SD) = (4800-4500)/(780) = (300)/(780) = 0.3846`

Now we have to find probabilities for both z-scores using the z-score table

`P(z_1<-1.92) =0.02743\\P(z_2<0.3846)= 0.6480`

Now for the probability between both values

`P(-1.92<z<0.3846) = P(z_2) - P(z_1) = 0.6480-0.02743 = 0.62057`

Rounding off to four decimal places

0.6206

Hence,

The probability that between 3,000 and 4,800 acres will be burned in any given year is 0.6206