A new roller coaster at an amusement park requires individua

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 mean=54.5 and Standard deviation =4.5

a. 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. A smaller coaster has a height requirement of 50 inches to ride. What proportion of 10 year-old-boys is tall enough to ride this coaster? Provide the probability statement (ie P(…)), show work, and value to 4 decimal places.

c.  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. 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

a)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    56      
u = mean =    54.5      
          
s = standard deviation =    4.5      
          
Thus,          
          
z = (x - u) / s =    0.333333333      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(x > 56) =    0.36944134 [ANSWER]

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B)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    50      
u = mean =    54.5      
          
s = standard deviation =    4.5      
          
Thus,          
          
z = (x - u) / s =    -1      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(x>50) =    0.841344746 [ANSWER]

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c)

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    50      
x2 = upper bound =    56      
u = mean =    54.5      
          
s = standard deviation =    4.5      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -1      
z2 = upper z score = (x2 - u) / s =    0.333333333      
          
Using table/technology, the left tailed areas between these z scores is          
          
          
P(z < z2) =    0.63055866      
          
Thus, the area between them, by subtracting these areas, is          
          
P(50<x<56) =    0.63055866   [ANSWER]

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D)

First, we get the z score from the given left tailed area. As          
          
Left tailed area =    0.25      
          
Then, using table or technology,          
          
z =    -0.67448975      
          
As x = u + z * s,          
          
where          
          
u = mean =    54.5      
z = the critical z score =    -0.67448975      
s = standard deviation =    4.5      
          
Then          
          
x = critical value =    51.46479612      

Thus, the height requirement is at least 51.46 in. [ANSWER]