To find the mean of the test scores, let's use the information provided about how the scores relate to the mean and the standard deviation.
1. According to the given information, a score of 43 is one standard deviation above the mean. We can express this mathematically as:
\[
\text{mean} + 1 \times \text{standard deviation} = 43
\]
2. Similarly, a score of 19 is three standard deviations below the mean:
\[
\text{mean} - 3 \times \text{standard deviation} = 19
\]
Let's denote:
- The mean of the test scores as \( m \)
- The standard deviation as \( \text{std} \)
Now we can rewrite our two equations as follows:
\[
m + \text{std} = 43 \quad \text{(Equation 1)}
\]
\[
m - 3\text{std} = 19 \quad \text{(Equation 2)}
\]
Our goal is to solve for \( m \). We can do this by using these two equations to express \( \text{std} \) in terms of \( m \) and then substituting it back.
From Equation 1:
\[
\text{std} = 43 - m
\]
Now, substitute \( \text{std} \) from Equation 1 into Equation 2:
\[
m - 3(43 - m) = 19
\]
To solve for \( m \), expand the equation:
\[
m - 129 + 3m = 19
\]
Combine like terms:
\[
4m - 129 = 19
\]
Add 129 to both sides to isolate the term with \( m \):
\[
4m = 19 + 129
\]
\[
4m = 148
\]
Finally, divide both sides by 4 to solve for \( m \):
\[
m = \frac{148}{4}
\]
\[
m = 37
\]
Therefore, the mean of the test scores is \( 37 \).
