Question

The widths of the heads of fireworks on Dragon Island M= 91mm and standard deviation=23mm and have a normal distribution.

a. By law, fireworms in the smallest 10th percentile must be released if captured. Find the head width that represents the 10th percentile.

b. It appears only fireworks with head widths of 102mm to 119mm make reliable circus performers. What percentage of fireworks does this represent ?

c. Offers a reward for the largest 10% of fireworks on the island. What head width represents the 90th percentile ?

Answer 1

Solution :

mean = = 91mm

standard deviation = = 23mm

Using standard normal table,

a ) P(Z < z) = 10%

P(Z < z) = 0.10

P(Z <-1.282) = 0.10

z = 1.28

Using z-score formula,

x = z * +

x = 1.28 * 23+ 91

= 61.56

The 10^th percentile. = 61.56

b ) P ( 102 < z < 119 )

P ( 102 - 91 / 23) < (Z -  / ) < ( 119 - 91 / 23)

P ( 11 / 23 < z < 28 / 23)

P (0.48 < z < 1.21 )

P (z < 1.21) - p ( z < 0.48 )

Using z table

= 0.8869- 0.6844

= 0.2025

Probability = 0.2025 = 20.25%

c ) P(Z < z) = 90%

P(Z < z) = 0.90

P(Z <- 1.282) = 0.90

z = 1.28

Using z-score formula,

x = z * +

x =-1.28 * 23+ 91

= 120.44

The 90^th percentile. = 120.44