Question

this is the only question in this section.
question
on a test that has a normal distribution, a score of 63 falls three standard deviations below the mean, and a score of 73 falls two standard deviations above the mean. determine the mean of this test.
answer
attempt 1 out of 2

Explanation:

Step1: Define variables for mean/std dev

Let $\mu$ = mean, $\sigma$ = standard deviation.
Translate the problem to equations:
$63 = \mu - 3\sigma$
$73 = \mu + 2\sigma$

Step2: Subtract equations to solve for $\sigma$

Subtract the first equation from the second:
$$73 - 63 = (\mu + 2\sigma) - (\mu - 3\sigma)$$
$$10 = 5\sigma$$
$$\sigma = 2$$

Step3: Substitute $\sigma$ to find $\mu$

Plug $\sigma=2$ into $73 = \mu + 2\sigma$:
$$73 = \mu + 2(2)$$
$$73 = \mu + 4$$
$$\mu = 73 - 4 = 69$$

Answer:

69