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The problem in the image involves calculating the probability that a sample of 35 exam scores will have a mean score of 528 or more, assuming the exam scores follow a normal distribution.

### Given dataThe problem in the image involves calculating the probability that a sample of 35 exam scores will have a mean score of 528 or more, assuming the exam scores follow a normal distribution.

### Given 
- Population mean (\(\mu\)) = 500
- Population standard deviation (\(\sigma\)) = 170
- Sample size (\(n\)) = 35
- Sample mean (\(\bar{x}\)) = 528

### Approach:
1. **Standard Error (SE)** of the sample mean is calculated using:
   \[
   SE = \frac{\sigma}{\sqrt{n}}
   \]

2. **Z-score** is calculated as:
   \[
   Z = \frac{\bar{x} - \mu}{SE}
   \]

3. **Probability**: Once the Z-score is calculated, we can look up the corresponding probability using the standard normal distribution table (or use a calculator for cumulative probability).

Let's calculate this step by step.Here are the calculations:

1. **Standard Error (SE)**: \( 28.735 \)
2. **Z-score**: \( 0.974 \)
3. **Probability**: \( 0.165 \) (or about 16.5%)

Thus, the probability that a sample of 35 exams will have a mean score of 528 or more, assuming the exam scores follow the same distribution as in the past, is approximately **16.5%**.

Would you like more details, or do you have any questions? Here are five related questions for further exploration:

1. How would the probability change if the sample size were larger, say 50?
2. What if the standard deviation were smaller? How would that affect the Z-score?
3. Could we use the empirical rule to approximate the probability in this case?
4. How is the Z-score related to hypothesis testing for sample means?
5. What assumptions must hold for the normal distribution to be valid in this case?

**Tip:** When working with normal distributions, always remember that a larger sample size will decrease the standard error, leading to more precise estimates of the population mean.

What is the probability that a sample of 35 exams will have a mean score of 528 or more, if the exam scores follow the same distribution as in the past?

Calculate the probability that a sample of 35 exams will have a mean score of 528 or more, assuming a population mean of 500 and a standard deviation of 170. Explore key statistical concepts such as standard error, Z-scores, and the normal distribution.

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