On a test that has a normal distribution, a score of 59 falls one standard deviation above the mean, and a score of 77 falls three standard deviations above the mean. Determine the mean of this test.
To determine the mean of the test, we need to first calculate the standard deviation.
Since a score of 59 falls one standard deviation above the mean, we can write this as:
59 = mean + 1 * standard deviation
Similarly, since a score of 77 falls three standard deviations above the mean, we can write this as:
77 = mean + 3 * standard deviation
Now we have a system of two equations with two unknowns (mean and standard deviation):
59 = mean + standard deviation
77 = mean + 3 * standard deviation
Subtracting the first equation from the second gives:
77 - 59 = mean + 3 * standard deviation - mean - standard deviation
18 = 2 * standard deviation
Dividing both sides by 2 gives:
standard deviation = 9
Now we can substitute the value of standard deviation back into the first equation to solve for the mean:
59 = mean + 9
mean = 59 - 9
mean = 50
Therefore, the mean of this test is 50.