Question

On a test that has a normal distribution, a score of 19 falls one standard deviation below the mean, and a score of 40 falls two standard deviations above the mean. Determine the mean of this test.

Answer 1

Hi there!

We can use the following equation:

`\large\boxed{z = (x-\mu)/(\sigma)}`

z = amount of standard deviations away a value is from the mean (z-score)

σ = standard deviation

x = value

μ = mean

Plug in the knowns for both and rearrange to solve for the mean:

`-1 = (19-\mu)/(\sigma)\\\\-\sigma = 19 - \mu\\\\ \sigma = -19 + \mu`

Other given:

`2 = (40-\mu)/(\sigma)\\\\2\sigma = 40- \mu\\\\ \sigma = 20 - (\mu)/(2)`

Set both equal to each other and solve:

`-19 + \mu = 20 - (\mu)/(2)\\\\(3\mu)/(2) = 39 \\\\3\mu = 78 \\\\\boxed{\mu = 26 }`