Question () Watch Video Show Examples On a test that has a normal distribution, a score of 31 falls two standard deviations below the mean, and a score of 71 falls three standard deviations above the mean. Determine the mean of this test. Answer Attempt 1 out of 2 Submit Answer

Step 1 ：The problem is asking for the mean of a normal distribution given two data points and their respective standard deviations from the mean.

Step 2 ：The formula for a z-score is \(Z = \frac{X - \mu}{\sigma}\), where \(Z\) is the z-score, \(X\) is the data point, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Step 3 ：We can rearrange this formula to solve for the mean: \(\mu = X - Z\sigma\).

Step 4 ：We have two equations with two unknowns (\(\mu\) and \(\sigma\)), so we can solve this system of equations to find the mean.

Step 5 ：Let's denote the two given data points and their z-scores as \((X1, Z1) = (31, -2)\) and \((X2, Z2) = (71, 3)\).

Step 6 ：We can set up the following system of equations: \(1) \mu = X1 - Z1\sigma\) and \(2) \mu = X2 - Z2\sigma\).

Step 7 ：We can solve this system of equations by setting the two expressions for \(\mu\) equal to each other and solving for \(\sigma\). Then we can substitute \(\sigma\) into one of the equations to find \(\mu\).

Step 8 ：The solution to the system of equations is \(\mu = 47\) and \(\sigma = 8\). This means that the mean of the test scores is 47 and the standard deviation is 8. This is consistent with the information given in the problem, as a score of 31 is indeed two standard deviations below the mean of 47, and a score of 71 is three standard deviations above the mean.

Step 9 ：Final Answer: The mean of the test scores is \(\boxed{47}\).