On a test that has a normal distribution, a score of 58 falls one standard deviation above the mean, and a score of 64 falls three standard deviations above the mean. Determine the mean of this test.
To determine the mean of the test, we first need to find the standard deviation.
Let's denote the mean as μ and the standard deviation as σ.
Given that a score of 58 falls one standard deviation above the mean, we can write this as:
58 = μ + σ
Similarly, a score of 64 falls three standard deviations above the mean:
64 = μ + 3σ
Now, we can solve these two equations simultaneously to find the values of μ and σ.
First, we can rewrite the first equation as:
μ = 58 - σ
Substitute this expression for μ into the second equation:
64 = (58 - σ) + 3σ
64 = 58 + 2σ
6 = 2σ
σ = 3
Now that we have found the value of the standard deviation to be 3, we can substitute this back into the first equation to find the mean:
58 = μ + 3
μ = 58 - 3
μ = 55
Therefore, the mean of this test is 55.