To find the probability that a randomly selected SAT score exceeds 1200, we need to use the properties of the normal distribution.
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Define the parameters:
- Mean (\(\mu\)) = 1028
- Standard Deviation (\(\sigma\)) = 92
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Standardize the score: We will convert the score of 1200 into a z-score using the formula: \[ z = \frac{X - \mu}{\sigma} \] where \(X\) is the score of interest.
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Calculate the z-score: \[ z = \frac{1200 - 1028}{92} = \frac{172}{92} \approx 1.8696 \]
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Use the z-score to find the probability: We now need to find the probability that a z-score is greater than 1.8696. This can be found using the standard normal distribution table, or by using a calculator or statistical software.
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Find the probability for \(z\):
- We need to find \(P(Z > 1.8696)\).
- First, we find \(P(Z < 1.8696)\). Using the standard normal distribution table or a calculator, we find that: \[ P(Z < 1.8696) \approx 0.9699 \]
- To find \(P(Z > 1.8696)\): \[ P(Z > 1.8696) = 1 - P(Z < 1.8696) = 1 - 0.9699 = 0.0301 \]
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Round the result: To express the probability as a decimal rounded to the nearest hundredth: \[ P(Z > 1.8696) \approx 0.0301 \approx 0.03 \]
Thus, the probability that a randomly selected score exceeds 1200 is \( \boxed{0.03} \).