Questions: In a particular year, the mean score on the ACT test was 20.9 and the standard deviation was 4.6 • The mean score on the SAT mathematics test was 502 and the standard deviation was 112 • The distributions of both scores were approximately bell-shaped. Round the answers to at least two decimal places. (d) Emma's SAT score had a z-score of -2.5 • What was her SAT score? Emma's SAT score is

Solution Steps

To find Emma's SAT score given her z-score, we can use the formula for the z-score: \( z = \frac{(X - \mu)}{\sigma} \), where \( X \) is the score, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. Rearrange this formula to solve for \( X \).

We need to find Emma's SAT score given her z-score of \(-2.5\). The mean SAT score is \( \mu = 502 \) and the standard deviation is \( \sigma = 112 \).

The z-score formula is given by: \[ z = \frac{(X - \mu)}{\sigma} \] where \( X \) is Emma's SAT score. We need to rearrange this formula to solve for \( X \).

Rearrange the z-score formula to solve for \( X \): \[ X = z \cdot \sigma + \mu \]

Substitute the known values into the rearranged formula: \[ X = (-2.5) \cdot 112 + 502 \]

Perform the calculation: \[ X = -280 + 502 = 222 \]

Final Answer