High School

Standardized tests: In a particular year, the mean score on the ACT test was 17.5, and the standard deviation was 4.8. The mean score on the SAT mathematics test was 549, and the standard deviation was 120. The distributions of both scores were approximately bell-shaped. Round the answers to at least two decimal places.

(e) Emma's SAT score had a z-score of -2.1. What was her SAT score?

Answer :

To find Emma's SAT score given her z-score, we can use the z-score formula:

[tex]\text{z-score} = \frac{(X - \mu)}{\sigma}[/tex]

where:

  • [tex]X[/tex] is the SAT score we are solving for,
  • [tex]\mu[/tex] is the mean SAT score, which is 549,
  • [tex]\sigma[/tex] is the standard deviation of SAT scores, which is 120,
  • [tex]\text{z-score}[/tex] is given as -2.1.

Rearrange the formula to solve for [tex]X[/tex]:

[tex]X = \mu + (\text{z-score} \times \sigma)[/tex]

Substitute the provided values into the equation:

[tex]X = 549 + (-2.1 \times 120)[/tex]

First, calculate the product:

[tex]-2.1 \times 120 = -252[/tex]

Then, substitute back to find [tex]X[/tex]:

[tex]X = 549 - 252 = 297[/tex]

Thus, Emma's SAT score was 297. This calculation shows how z-scores can be used to determine individual data points from a normal distribution, using the mean and standard deviation.

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