The distribution of scores on the SAT is approximately normal with a mean of mu = 500 and a standard deviation of sigma = 100. For the population of students who have taken the SAT, a. What proportion have SAT scores greater than 700? b. What proportion have SAT scores greater than 550? c. What is the minimum SAT score needed to be in the highest 10% of the population? d. If the state college only accepts students from the top 60% of the SAT distribution, what is the minimum SAT score needed to be accepted?

a. 2.28%

b. 30.85%

c. 628.16

d. 474.67

Step-by-step explanation:

For a given value x, the related z-score is computed as z = (x-500) / 100.

a. The z-score related to 700 is (700-500) / 100 = 2, and P (Z > 2) = 0.0228 (2.28%)

b. The z-score related to 550 is (550-500) / 100 = 0.5, and P (Z > 0.5) = 0.3085 (30.85%)

c. We are looking for a value b such that P (Z > b) = 0.1, i. e., b is the 90th quantile of the standard normal distribution, so, b = 1.281552. Therefore, P ((X-500) / 100 > 1.281552) = 0.1, equivalently P (X > 500 + 100 (1.281552)) = 0.1 and the minimun SAT score needed to be in the highest 10% of the population is 628.1552

d. We are looking for a value c such that P (Z > c) = 0.6, i. e., c is the 40th quantile of the standard normal distribution, so, c = - 0.2533471. Therefore, P ((X-500) / 100 > - 0.2533471) = 0.6, equivalently P (X > 500 + 100 (-0.2533471)), and the minimun SAT score needed to be accepted is 474.6653