Question

In Example 2-1, part c, the data were represented by the normal distribution function f(x)=0.178 exp(-0.100(x-451)2 Use this distribution function to determine the fraction of individuals demonstrating a response in the range of 2.5 to 7.5.

Answer 1

P ( 2.5 < X < 7.5 ) = 0.7251

Step-by-step explanation:

Given:

- The pmf for normal distribution for random variable x is given:

f(x)=0.178 exp(-0.100(x-4.51)^2)

Find:

the fraction of individuals demonstrating a response in the range of 2.5 to 7.5.

Solution:

- The random variable X follows a normal distribution with mean u = 4.51, and standard deviation s.d as follows:

s.d = sqrt ( 1 / 0.1*2)

s.d = sqrt(5) =2.236067

- Hence, the normal distribution is as follows:

X ~ N(4.51 , 2.236)

- Compute the Z-score values of the end points 2.5 and 7.5:

P ( (2.5 - 4.51) / 2.236 < Z < (7.5 - 4.51 ) / 2.236 )

P ( -0.898899327 < Z < 1.337168651 )

- Use the Z-Table for the probability required:

P ( 2.5 < X < 7.5 ) = P ( -0.898899327 < Z < 1.337168651 ) = 0.7251