Step 1 :Find the z-scores for 606 and 1446 using the formula \(z = \frac{x - \mu}{\sigma}\), where \(x\) is the score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
Step 2 :Calculate the z-scores: \(z_1 = \frac{606 - 1026}{210} = -2.0\) and \(z_2 = \frac{1446 - 1026}{210} = 2.0\)
Step 3 :Use the standard normal distribution table to find the percentage of students who scored above 606: \(P(z > -2.0) = 0.9772\)
Step 4 :Calculate the percentage of students who scored between 606 and 1446: \(P(-2.0 < z < 2.0) = 0.9772 - 0.0228 = 0.9544\)
Step 5 :Find the number of students out of 500 who scored between 606 and 1446: \(500 \times 0.9544 = 477.2\)
Step 6 :\(\boxed{\text{(a) 97.7\% of students scored over 606.}}\)
Step 7 :\(\boxed{\text{(b) We expect about 477 students to have SAT scores between 606 and 1446 out of every 500 students.}}\)