Question

Scores on the SAT critical reading test in 2015 follow a Normal distribution with a mean of 495 and a standard deviation of 116

a. What proportion of students who took the SAT critical reading test had scores above 600?

b. What proportion of students who took the SAT critical reading test had scores between 400 and 600?

c. Jacob took the SAT critical reading test in 2015 and scored a 640. Janet took the ACT critical reading test which also follows a Normal distribution with a mean of 21 and a standard deviation of 5.5. Janet scored a 31 on the ACT critical reading test. Who did better?

Answer 1

a)

Here, μ = 495, σ = 116 and x = 600. We need to compute P(X >= 600). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (600 - 495)/116 = 0.91

Therefore,
P(X >= 600) = P(z <= (600 - 495)/116)
= P(z >= 0.91)
= 1 - 0.8186 = 0.1814

b)

Here, μ = 495, σ = 116, x1 = 400 and x2 = 600. We need to compute P(400<= X <= 600). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (400 - 495)/116 = -0.82
z2 = (600 - 495)/116 = 0.91

Therefore, we get
P(400 <= X <= 600) = P((600 - 495)/116) <= z <= (600 - 495)/116)
= P(-0.82 <= z <= 0.91) = P(z <= 0.91) - P(z <= -0.82)
= 0.8186 - 0.2061
= 0.6125

c)

For SAT

Here, μ = 495, σ = 116 and x = 640. We need to compute P(X >= 640). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (640 - 495)/116 = 1.25

For ACT

Here, μ = 21, σ = 5.5 and x = 31. We need to compute P(X >= 31). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (31 - 21)/5.5 = 1.82

Janet did better because z value is more than Jacob