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The question asks for which of the given shaded areas in the graphs represents the proportion of students who scored higher than Luis, who earned **1248** on the SAT. The SAT scores are normally distributed with a **mean of 1100** and **standard deviation of 200**.

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### **Solution Process:**

1. **Standardizing the score of 1248:**

   The **z-score** formula helps convert a score into the number of standard deviations above or below the mean:
   \[
   z = \frac{X - \mu}{\sigma}
   \]
   where:
   - \(X = 1248\) (Luis's score)
   - \(\mu = 1100\) (mean)
   - \(\sigma = 200\) (standard deviation)

   Substituting the values:
   \[
   z = \frac{1248 - 1100}{200} = \frac{148}{200} = 0.74
   \]

2. **Finding the proportion of students scoring higher than Luis:**

   Using a **z-table**, a z-score of **0.74** corresponds to a **cumulative area** of approximately **0.7704**. This means 77.04% of the scores are **below** 1248.

   Thus, the **proportion of students scoring higher** than Luis is:
   \[
   1 - 0.7704 = 0.2296 \approx 23\%
   \]

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### **Selecting the Correct Graph:**

- We need to find the graph that shows the **right tail** shaded (since we are looking for scores higher than 1248).

- By observing the images:
  - **First graph:** Shows a right tail shaded, starting around **1300**.
  - **Second graph:** Shows a left-side area (incorrect for "higher than" Luis).
  - **Third graph:** Shows the middle part shaded (also incorrect for scores above 1248).

Thus, the **correct graph is the first one**, as it represents the area to the **right** of a value near Luis’s score (1248). This shaded region corresponds to students with **higher scores** than him.

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Would you like further explanation on z-scores, normal distribution, or how to interpret z-tables?

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Here are 5 related questions you might explore:
1. How do z-scores relate to probability?
2. How can you calculate probabilities for any z-score using a z-table?
3. What is the central limit theorem, and how does it affect distributions?
4. How do standard deviations affect the spread of a normal distribution?
5. What proportion of students fall within 1 standard deviation of the mean?

**Tip:** When working with normal distributions, always visualize whether the question asks for a left-tail, right-tail, or central probability to choose the correct area!

Suppose scores on the SAT exam are normally distributed with mean 1100 and standard deviation 200. Luis scored 1248 on the SAT. Which of the shaded areas in the following graphs represent the proportion of students who earned a higher score than Luis?

Learn how to find the proportion of students scoring higher than 1248 on the SAT using normal distribution properties. The SAT scores are normally distributed with a mean of 1100 and a standard deviation of 200. Understand z-scores and statistical methods for solving this problem.

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