Suppose SAT scores are normally distributed with a mean of 1000 (out of 1600) an

Question

Suppose SAT scores are normally distributed with a mean of 1000 (out of 1600) and a standard deviation of 200.

What must your score be (at least) if you are in the top 5% of students?

Suppose the SAT offers certificates of excellence to any student with a score above 1250. If you know that a randomly-chosen student has been awarded a certificate of excellence, what is the probability the student has a score above 1400?

If you pick students randomly and one-at-a-time, how many will you have to pick before finding a student with a score below 700?

Explanation / Answer

Mean = 1000

Standard deviation = 200

For top 5%,

P(X < A) = 0.95

P(Z < (A - mean)/standard deviation) = 0.95

P(Z < (A - 1000)/200) = 0.95

(A - 1000)/200 = 1.645

A = 1329

SAT score must be at least 1329 if you are in the top 5% of students

P(1250 < X < 1400) = P(X < 1600) - P(X < 1250)

= P(Z <4) - P(Z < (1250-1000)/200)

= 1 - P(Z < 1.25)

= 1 - 0.8944

= 0.1056

P(1400 < X < 1600) = P(X < 1600) - P(X<1400)

= P(Z < 4) - P(Z < (1400-1000)/200)

= 1 - P(Z < 2)

= 1 - 0.9772

= 0.0228

If you know that a randomly-chosen student has been awarded a certificate of excellence, the probability the student has a score above 1400 = P(X>1400)/P(X>1250)

= 0.0228/0.1056 = 0.2159

Probability of a student having score below 700 = P(Z < 700-1000)/200)

= P(Z < -1.5)

= 0.0668

Number of people you have to pick before finding a student with a score below 700 = 1/0.0668 = 15

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