Question 1140214
assuming the distribution of the scores are normal, then you get the following.


z = (x - m) / s


z is the z-score
x is the raw score
m is the mean
s is the standard deviation


if the mean is 12.8 and the high score is 16.4, then the formula becomes:


z = 16.4 - 12.8 / s which becomes z = 3.6 / s


if you solve for s, you will get s = 3.6 / z


you need to find z.


you are given that 77% of the scores are between 9.2 and 16.4.


the 77% is assumed to be in the middle of the normal distribution curve.


this means that the tails on either end are the same.


each tail is therefore (100% - 77%) / 2 = 11.5%.


if you look up the z-score that has 11.5% of the normal distribution curve to the left of it, you will find that the z-score is equal to -1.200358858.


that would be the low z-score.
since the normal distribution is symmetric, then the high z-score will be 1.200358858.


you have 77% of the normal distribution is between z-score of -1.200358858. and 1.200358858.


your high z-score formula becomes z = (16.4 - 12.8) / s which becomes 1.200358858 = 3.6 / s.


solve for s to get s = 3.6/1.200358858 which makes s = 2.999103123


that would be your standard deviation.


your mean is 12.8
your standard deviation is 2.999103123
77% of your scores will be between 9.2 and 16.4