Question

The average length of a certain species of snake is known to be 60.96 inches (σ = 5.31 inches), and their length follows a Normal distribution.

Write the probability statement, and show your steps in calculating your answer

40% of the population is smaller than what length?

75% of the population is between 50inches and how many inches?

What percent of the population is between 55 and 65 inches?

What percent of the population is larger than 65 inches?

Answer 1

mean = 60.96, s = 5.31

a)

z value at 40% = -0.253
by using z standard left tailed table,

z = (x - mean)/s

-0.2533 = (x - 60.96)/5.31
x = 59.6147

b)

P(50 < x < a ) = 0.75

P(X <a) - P(x< 50) =0.75

0.75 = P(x<a) - P(x< 50)
= 0.75 + P(z < (50 - 60.96)/5.31)
= 0.75 + P(z < -2.0640)
= 0.75 + 0.0195
= 0.7695

P(x<a) = 0.7695
z value will be positive, because it llies above 0.5
So, z = 0.737
By using central limit theorem,

z = (x - mean)/s
0.737 = (x - 60.96)/5.31
x = 64.8735

P(50 < x < 64.8735) = 0.75

c)

P(55< x< 65)
= P((55 - 60.96)/5.31 <z < (65 - 60.96)/5.31 )
= P(-1.1224 < z < 0.7608)
= P(z< 0.7608) - P(z< -1.1224)
= 0.6458
= 64.58%

d)

P(x > 65)
= P(z > (65 - 60.96)/5.31)
= P(z> 0.7608)
= 1 - P(z< 0.7608)
= 1 - 0.7766
= 0.2234 = 22.34%