Question

The percent of persons (ages five and older) in each state who speak a language at home other than English is approximately exponentially distributed with a mean of 8.76.

The lambda of this distribution is

The probability that the percent is larger than 3.24 is P(x ≥ 3.24) =

The probability that the percent is less than 9.79 is P(x ≤ 9.79) =

The probability that the percent is between 5.76 and 11.76 is P(5.76 ≤ x ≤ 11.76) =

Answer 1

Let X be the percentage of persons who speak other language       
X follows exponential distribution with mean = 8.76       
Since mean = 8.76       

1) λ = 0.1142       
The lambda of this distribution is 0.1142       
        
We can find the probabilities substituting X value in the CDF of the Exponential distribution       
CDF of the Exponential distribution is        

2) P(X ≥ 3.24) = 1 - P(X < 3.24)       
   = 1 - F(3.24)      
   = 1 - 0.3093      
   = 0.6907      
The probability that the percent is larger than 3.24 is P(x ≥ 3.24) = 0.6907       
        
3) P(X ≤ 9.79) = F(9.79)       
   = 0.6731      
The probability that the percent is less than 9.79 is P(x ≤ 9.79) = 0.6731       
        
4) P(5.76 ≤ x ≤ 11.76) = P(X ≤ 11.76) - P(X ≤ 5.76)       
   = F(11.76) - F(5.76)      
   = 0.7389 - 0.4820      
   = 0.2569      
The probability that the percent is between 5.76 and 11.76 is P(5.76 ≤ x ≤ 11.76) = 0.2569