On a test that has normal distribution of scores, a score of 58 falls two standard deviation below the mean, and a score of 85 is one standard deviation above the mean. What is the mean score of this test?

Answer

To find the mean score of the test based on the given information, we can follow these steps:

### Step 1: Understand the relationship between scores, mean, and standard deviation.
We know that:
- A score of 58 is 2 standard deviations below the mean.
- A score of 85 is 1 standard deviation above the mean.

### Step 2: Set up equations based on the information provided.
Let the mean score be represented by \( \mu \) (mean) and the standard deviation by \( \sigma \) (standard deviation). We can express the scores in terms of the mean and standard deviation:
- For the score of 58:
\[ 58 = \mu - 2\sigma \]
- For the score of 85:
\[ 85 = \mu + \sigma \]

### Step 3: Solve the equations simultaneously.
We have two equations:
1. \( 58 = \mu - 2\sigma \) (Equation 1)
2. \( 85 = \mu + \sigma \) (Equation 2)

### Step 4: Rearrange Equation 1 to express \( \mu \) in terms of \( \sigma \).
From Equation 1:
\[ \mu = 58 + 2\sigma \]

### Step 5: Substitute \( \mu \) from Equation 1 into Equation 2.
Substituting \( \mu \) in Equation 2:
\[ 85 = (58 + 2\sigma) + \sigma \]
This simplifies to:
\[ 85 = 58 + 3\sigma \]

### Step 6: Solve for \( \sigma \).
Subtract 58 from both sides:
\[ 85 - 58 = 3\sigma \]
\[ 27 = 3\sigma \]
Now divide by 3:
\[ \sigma = 9 \]

### Step 7: Substitute \( \sigma \) back to find \( \mu \).
Now substitute \( \sigma \) back into the equation for \( \mu \):
\[ \mu = 58 + 2(9) \]
\[ \mu = 58 + 18 \]
\[ \mu = 76 \]

### Conclusion:
The mean score of the test is \( \mu = 76 \).

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