On a test that has a normal distribution, a score of 37 falls one standard deviation above the mean, and a score of 10 falls two standard deviations below the mean. Determine the mean of this test.
To find the mean of the test, we first need to identify the standard deviation.
Since a score of 37 falls one standard deviation above the mean, we know that:
37 = mean + 1*standard deviation
Similarly, since a score of 10 falls two standard deviations below the mean, we have:
10 = mean - 2*standard deviation
Now we can set up a system of equations:
37 = mean + standard deviation
10 = mean - 2*standard deviation
Next, we can solve for the mean by adding the two equations together:
37 + 10 = mean + standard deviation + mean - 2*standard deviation
47 = 2*mean - 1*standard deviation
Now, we can substitute back into one of the original equations to solve for the mean:
37 = mean + standard deviation
37 = (47 + standard deviation)/2 + standard deviation
37 = 47/2 + 1.5*standard deviation
74 = 47 + 3*standard deviation
3*standard deviation = 27
standard deviation = 9
Finally, substitute the standard deviation back into one of the original equations to solve for the mean:
37 = mean + 9
mean = 37 - 9
mean = 28
Therefore, the mean of the test is 28.