6 A new roller coaster at an amusement park requires individ

6. A new roller coaster at an amusement park requires individuals to be at least 4’8” (56 inches) tall to ride. It is estimated that the heights of 10 –year-old boys are normally distributed with = 54.5 and = 4.5 . a. (2pts) What proportion of 10-year-old boys is tall enough to ride the coaster? Provide the probability statement (ie P(…)), show work, and value to 4 decimal places. b. (2pts) A smaller coaster has a height requirement of 50 inches to ride. What proportion of 10 year-oldboys is tall enough to ride this coaster? Provide the probability statement (ie P(…)), show work, and value to 4 decimal places. c. (2pts) What proportion of 10-year old boys is tall enough to ride the coaster in part b but not tall enough to ride the coaster in part a? Provide the probability statement (ie P(…)), show work, and value to 4 decimal places. d. (4pts) The amusement park wants to create a roller coaster so that 75% of 10-year-old boys can ride the coaster. What should the height requirement be set at? Show work.

Solution

Normal Distribution
Mean ( u ) =54.5
Standard Deviation ( sd )=4.5
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X < 56) = (56-54.5)/4.5
= 1.5/4.5= 0.3333
= P ( Z <0.3333) From Standard Normal Table
= 0.6306                  
P(X > = 56) = (1 - P(X < 56)
= 1 - 0.6306 = 0.3694 ~ 36.94%                  

b)
P(X < 50) = (50-54.5)/4.5
= -4.5/4.5= -1
= P ( Z <-1) From Standard Normal Table
= 0.1587                  
P(X > = 50) = (1 - P(X < 50)
= 1 - 0.1587 = 0.8413   ~ 84.13%              

d)
P ( Z < x ) = 0.25
Value of z to the cumulative probability of 0.25 from normal table is -0.674
P( x-u/s.d < x - 54.5/4.5 ) = 0.25
That is, ( x - 54.5/4.5 ) = -0.67
--> x = -0.67 * 4.5 + 54.5 = 51.467 is the height measure they consider