Question

the scores on a test are normally distributed with a mean of 130 and a standard deviation of 26. what is the score that is 3 standard deviations below the mean? a score of □ is 3 standard deviations below the mean.

Explanation:

Step1: Recall the formula for z - score

The formula for z - score is \(z=\frac{x-\mu}{\sigma}\), where \(x\) is the raw score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. We are given that \(z=- 3\) (since it is 3 standard deviations below the mean), \(\mu = 130\), and \(\sigma=26\).

Step2: Substitute the values into the formula and solve for \(x\)

Substitute \(z=-3\), \(\mu = 130\), and \(\sigma = 26\) into \(z=\frac{x-\mu}{\sigma}\).
We get \(-3=\frac{x - 130}{26}\).
Multiply both sides by 26: \(-3\times26=x - 130\).
\(-78=x - 130\).
Add 130 to both sides: \(x=130-78\).

Answer:

\(52\)