SAT scores were originally created lo be normally distributed, with mu=500 and s

Question

SAT scores were originally created lo be normally distributed, with mu=500 and sigma=100. (a) What score marks the top 7 percent of the population? (b) What score marks the bottom 15 percent of the population? (c) What scores mark the middle 99 percent of the population? The total loud on a building is modeled as the sum of die dead load and occupancy load, l he dead load is modeled as a gamma distribution with mean 100 and variance 200 while the occupancy load is modeled its a gamma with mean 60 and variance 120. Find the mean, variance and p.d.f. of the total sustained load on the building

Explanation / Answer

Mean = 500

Standard deviation = 100

A) Top 7% mean P(Z<X) =0.93

P(z < (X - mean)/standard deviation) = 0.93

(X-500)/100 = 1.48

X = 648

A score of 648 marks the top 7%

B) For bottom 15%,

P(Z<X) = 0.15

(X-500)/100 = -1.04

X = 396

A score of 396 marks the bottom 15%

C) Middle 99% means scores between P(Z<X1) = 0.005 and P(Z<X2) = 0.995

(X1 - 500)/100 = -2.575 and (X2 - 500)/100 = 2.575

X1 = 242.5 and X2 = 757.5

Scores of 22.5 and 757.5 marks the middle 99%

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