Question

Heights for a certain group of people are normally distributed with mean = 67 inches and standard deviation = 2.3 inches. Find the proportion of people in the group whose heights fall into the following ranges. (Round your answers to four decimal places.)

(a) Between 64 inches and 67 inches.


(b) Between 62 inches and 72 inches.


(c) Less than 72 inches.


(d) Greater than 62 inches.


(e) Either less than 62 inches or greater than 72 inches.

Answer 1

a)

Here, μ = 67, σ = 2.3, x1 = 64 and x2 = 67. We need to compute P(64<= X <= 67). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (64 - 67)/2.3 = -1.3
z2 = (67 - 67)/2.3 = 0

Therefore, we get
P(64 <= X <= 67) = P((67 - 67)/2.3) <= z <= (67 - 67)/2.3)
= P(-1.3 <= z <= 0) = P(z <= 0) - P(z <= -1.3)
= 0.5 - 0.0968
= 0.4032

b)
Here, μ = 67, σ = 2.3, x1 = 62 and x2 = 72. We need to compute P(62<= X <= 72). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (62 - 67)/2.3 = -2.17
z2 = (72 - 67)/2.3 = 2.17

Therefore, we get
P(62 <= X <= 72) = P((72 - 67)/2.3) <= z <= (72 - 67)/2.3)
= P(-2.17 <= z <= 2.17) = P(z <= 2.17) - P(z <= -2.17)
= 0.985 - 0.015
= 0.9700

c)

Here, μ = 67, σ = 2.3 and x = 72. We need to compute P(X <= 72). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (72 - 67)/2.3 = 2.17

Therefore,
P(X <= 72) = P(z <= (72 - 67)/2.3)
= P(z <= 2.17)
= 0.9850

d)

Here, μ = 67, σ = 2.3 and x = 62. We need to compute P(X >= 62). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (62 - 67)/2.3 = -2.17

Therefore,
P(X >= 62) = P(z <= (62 - 67)/2.3)
= P(z >= -2.17)
= 1 - 0.015 = 0.9850

e)

Here, μ = 67, σ = 2.3 and x = 62. We need to compute P(X <= 62). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (62 - 67)/2.3 = -2.17

Therefore,
P(X <= 62) = P(z <= (62 - 67)/2.3)
= P(z <= -2.17)
= 0.015

Here, μ = 67, σ = 2.3 and x = 72. We need to compute P(X >= 72). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (72 - 67)/2.3 = 2.17

Therefore,
P(X >= 72) = P(z <= (72 - 67)/2.3)
= P(z >= 2.17)
= 1 - 0.985 = 0.015

P(Either less than 62 inches or greater than 72 inches. ) = 0.0150 + 0.0150 = 0.0300