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Answer :
To solve this problem, we need to compare the two scores (Regan's SAT score and Veronica's ACT score) in terms of their z-scores. A z-score represents how many standard deviations a score is away from the mean. Once we find Regan's z-score on the SAT, we can determine the ACT score that would give Veronica the same z-score.
Step 1: Calculate Regan's z-score for the SAT
The formula for the z-score is:
\[
z = \frac{x - \mu}{\sigma}
\]
where:
- \(x\) is the score,
- \(\mu\) is the mean, and
- \(\sigma\) is the standard deviation.
For the SAT:
- \(x = 754\),
- \(\mu = 518\),
- \(\sigma = 118\).
Now, calculate Regan's z-score:
\[
z = \frac{754 - 518}{118} = \frac{236}{118} = 2.0
\]
Step 2: Determine the ACT score corresponding to the same z-score
Now, we want to find the ACT score that gives Veronica the same z-score of 2.0. For the ACT:
- The mean is \(\mu = 20.7\),
- The standard deviation is \(\sigma = 5\),
- We are solving for \(x\) (the score).
Using the z-score formula again:
\[
z = \frac{x - \mu}{\sigma}
\]
Substitute \(z = 2.0\), \(\mu = 20.7\), and \(\sigma = 5\):
\[
2.0 = \frac{x - 20.7}{5}
\]
Now, solve for \(x\):
\[
x - 20.7 = 2.0 \times 5 = 10
\]
\[
x = 10 + 20.7 = 30.7
\]
Final Answer:
Veronica would need to score 30.7 on the ACT to meet or beat Regan's score.
The correct answer is D. 30.7.
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