The GRE Scores are represented as ~N(310,12)
In order to find the proportion of scores between 286 and 322, we need to standardize the scores so we can use the standard normal probabilities. Thus, we will find the z-score.
[tex]z-score = \frac{286 - 310}{12} = -2[/tex]
[tex]z-score = \frac{322 - 310}{12} = 1 [/tex]
By looking on the standard normal probabilities table, we find the proportion of scores less than -2.
P(z < -2) = 0.0228
Then, we find the proportion of scores less than 1.
P(z < 1) = 0.8413
To find the proportion between -2 and 1, we subtract the two.
P(-2 < z < 1) = 0.8413 - 0.0228 = 0.8185 = 81.85%
Therefore, 82% of scores are between 286 and 322
