Question

Heights of 10 year olds, regardless of gender, closely follow a normal distribution with mean 55 inches and standard deviation 6 inches.

a) What is the probability that a randomly chosen 10 year old is shorter than 48 inches? (please respond to 2 decimal places, or use exact fractions)
answer:

b) What is the probability that a randomly chosen 10 year old is between 60 and 65 inches? (again, 2 decimal places, or use exact fractions)
answer:

c) If the tallest 10% of the class is considered "very tall", what is the height cutoff for "very tall"? (Yep, 2 decimal places, or use exact fractions)
answer:  inches

d) The height requirement for Batman the Ride at Six Flags Magic Mountain is 54 inches. What percent of 10 year olds cannot go on this ride? (Respond to 2 decimal places, do not include percent symbol)
answer:  %

Answer 1

a)

Here, μ = 55, σ = 6 and x = 48. We need to compute P(X <= 48). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (48 - 55)/6 = -1.17

Therefore,
P(X <= 48) = P(z <= (48 - 55)/6)
= P(z <= -1.17)
= 0.12

b)

Here, μ = 55, σ = 6, x1 = 60 and x2 = 65. We need to compute P(60<= X <= 65). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (60 - 55)/6 = 0.83
z2 = (65 - 55)/6 = 1.67

Therefore, we get
P(60 <= X <= 65) = P((65 - 55)/6) <= z <= (65 - 55)/6)
= P(0.83 <= z <= 1.67) = P(z <= 1.67) - P(z <= 0.83)
= 0.9525 - 0.7967
= 0.15

c)

z value at 10% = 1.28

z = (x - mean)/sigam

1.28 = (x - 55)/6

x = 6 *1.28 + 55
x = 62.68

d)
Here, μ = 55, σ = 6 and x = 54. We need to compute P(X <= 54). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (54 - 55)/6 = -0.17

Therefore,
P(X <= 54) = P(z <= (54 - 55)/6)
= P(z <= -0.17)
= 0.4325
= 43.25