Questions: Suppose SAT Writing scores are normally distributed with a mean of 489 and a standard deviation of 112. A university plans to award scholarships to students whose scores are in the top 4%. What is the minimum score required for the scholarship? Round your answer to the nearest whole number, if necessary.

Suppose SAT Writing scores are normally distributed with a mean of 489 and a standard deviation of 112. A university plans to award scholarships to students whose scores are in the top 4%. What is the minimum score required for the scholarship? Round your answer to the nearest whole number, if necessary.

Transcript text: Suppose SAT Writing scores are normally distributed with a mean of 489 and a standard deviation of 112. A university plans to award scholarships to students whose scores are in the top $4\%$. What is the minimum score required for the scholarship? Round your answer to the nearest whole number, if necessary.

Solution

Solution Steps

Step 1: Determine the Parameters

The SAT Writing scores are normally distributed with the following parameters:

Mean ($ \mu $): 489
Standard Deviation ($ \sigma $): 112

Step 2: Identify the Percentile

The university plans to award scholarships to students whose scores are in the top $ 4\% $. This corresponds to the $ 96\% $ percentile of the distribution.

Step 3: Find the Z-Score

To find the minimum score required for the scholarship, we first need the z-score that corresponds to the $ 96\% $ percentile. From standard normal distribution tables, the z-score for the $ 96\% $ percentile is approximately: \[ z \approx 1.7507 \]

Step 4: Calculate the Minimum Score

Using the z-score formula: \[ X = \mu + z \cdot \sigma \] we can substitute the values: \[ X = 489 + 1.7507 \cdot 112 \]

Calculating this gives: \[ X \approx 489 + 196.0004 \approx 685 \]