- x: the mean of the test
- s: the standard deviation of the test
We know that a score of 66 falls two standard deviations above the mean, so we can write:
66 = x + 2s
Similarly, we know that a score of 36 falls one standard deviation below the mean, so we can write:
36 = x - s
Now we have two equations with two unknowns (x and s). We can solve for x by isolating it in one of the equations and then substituting the result into the other equation.
Let's start with the second equation:
36 = x - s
x = 36 + s
Now we can substitute this expression for x into the first equation:
66 = x + 2s
66 = (36 + s) + 2s
66 = 36 + 3s
30 = 3s
s = 10
We have found the value of the standard deviation to be 10. Now we can substitute this value into either of the original equations to find the mean:
x = 36 + s
x = 36 + 10
x = 46
Therefore, the mean of the test is 46.