Question

The average LSAT score (the standardized test required to apply to law school) in the United States is µ =150 (σ = 10). Also, the LSAT is normally distributed. Use these parameters to answer the following questions:

• If someone took an LSAT test and received a 153, what proportion of scores will be less than this?
• If someone took an LSAT test and received a 143, what proportion of scores will be greater than this?
• What proportion of LSAT scores will be within the interval of 138 to 172?
• What proportion of LSAT scores will be outside the interval of 125 to 175?
• If someone wants to have an LSAT score higher than 90% of all other test-takers, what score do they need to earn?

Answer 1

a)

Here, μ = 150, σ = 10 and x = 153. We need to compute P(X <= 153). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (153 - 150)/10 = 0.3

Therefore,
P(X <= 153) = P(z <= (153 - 150)/10)
= P(z <= 0.3)
= 0.6179

b)
Here, μ = 150, σ = 10 and x = 143. We need to compute P(X >= 143). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (143 - 150)/10 = -0.7

Therefore,
P(X >= 143) = P(z <= (143 - 150)/10)
= P(z >= -0.7)
= 1 - 0.242 = 0.758

c)
Here, μ = 150, σ = 10, x1 = 138 and x2 = 172. We need to compute P(138<= X <= 172). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (138 - 150)/10 = -1.2
z2 = (172 - 150)/10 = 2.2

Therefore, we get
P(138 <= X <= 172) = P((172 - 150)/10) <= z <= (172 - 150)/10)
= P(-1.2 <= z <= 2.2) = P(z <= 2.2) - P(z <= -1.2)
= 0.9861 - 0.1151
= 0.8710

d)
Here, μ = 150, σ = 10, x1 = 125 and x2 = 175. We need to compute P(125<= X <= 175). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (125 - 150)/10 = -2.5
z2 = (175 - 150)/10 = 2.5

Therefore, we get
P(125 <= X <= 175) = P((175 - 150)/10) <= z <= (175 - 150)/10)
= P(-2.5 <= z <= 2.5) = P(z <= 2.5) - P(z <= -2.5)
= 0.9938 - 0.0062
= 0.9876
= 1 - 0.9876
= 0.0124

e)

z value at top 90% = 1.28

z = (x - mean)/sigma
1.28 = (x - 150)/10

x = 10 *1.28 + 150
x =162.80