Question

Suppose SAT reading scores are normally distributed with a mean of 496 and a standard deviation of 109. The University plans towards scholarships for students who scores are in the top 7%. What is the minimum score required for the scholarship round your answer to the nearest whole number.

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Sigrid

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To find the minimum score required for the top 7% of SAT reading scores, you can use the z-score formula. The z-score represents how many standard deviations a particular score is from the mean in a normal distribution.
The formula for the z-score is given by:\[ Z = \frac{(X - \mu)}{\sigma} \]
Where:
- X is the raw score,
- μ is the mean of the distribution,
- σ is the standard deviation.
In this case, you want to find the z-score corresponding to the top 7%, which means finding the z-score that leaves 7% in the tail. You can find this value using a standard normal distribution table or a calculator.
For the top 7%, you would look up the z-score that corresponds to the cumulative probability of 0.93 (since 100% - 7% = 93%). Using a standard normal distribution table or calculator, you find that \( Z \approx 1.44 \).
Now, plug this z-score back into the z-score formula to find the corresponding raw score (X):\[ 1.44 = \frac{(X - 496)}{109} \]
Solve for X :
\[ X = 1.44 \times 109 + 496 \]
\[ X = 642.96 \]
Rounding to the nearest whole number, the minimum score required for the scholarship is 643.

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