https://file.aiquickdraw.com/mathbot/file/1727742853247gepvn3ew.jpg

The problem is asking for the probability that the sample mean score is less than 484, given that:

- Population mean (\( \mu \)) = 495
- Population standard deviation (\( \sigma \)) = 111
- Sample size (\( n \)) = 61

To calculate this probability, we need to use the Central Limit Theorem, which states that the sampling distribution of the sample mean follows a normal distribution with:

- Mean \( \mu_{\bar{x}} = \mu = 495 \)
- Standard error \( \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{111}{\sqrt{61}} \)

### Steps:

1. **Calculate the standard error of the sample mean**:
   \[
   \sigma_{\bar{x}} = \frac{111}{\sqrt{61}} \approx 14.2076
   \]

2. **Find the z-score** for the sample mean \( \bar{x} = 484 \):
   \[
   z = \frac{484 - 495}{14.2076} \approx \frac{-11}{14.2076} \approx -0.774
   \]

3. **Look up the z-score** in the standard normal distribution table or use a calculator to find the probability corresponding to \( z = -0.774 \).

   The cumulative probability for \( z = -0.774 \) is approximately **0.2194**.

Therefore, the probability that the sample mean score is less than 484 is about **0.2194**, which matches the given answer in the screenshot.

Would you like a detailed explanation of any step? 

Here are some related questions you might explore:
1. How does increasing the sample size affect the standard error and the probability?
2. What is the probability that the sample mean score is greater than 505?
3. How would the calculation change if the sample size was smaller, say \( n = 30 \)?
4. What is the probability that the sample mean score is between 484 and 495?
5. How is the Central Limit Theorem applicable in real-world data sampling?

**Tip**: Always ensure the sample size is large enough for the Central Limit Theorem to hold, typically \( n \geq 30 \).

What is the probability that the sample mean score is less than 484? Assume that the population mean SAT score is 495, the standard deviation is 111, and the sample size is 61.

Learn how to calculate the probability that a sample mean SAT score is less than 484 using the Central Limit Theorem and z-scores. The population mean is 495, with a standard deviation of 111, and a sample size of 61. Step-by-step solution provided with key statistical concepts.

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