Find the probability that the mean of a sample of 25 people greater than 10

To find the probability that the mean of a sample of 25 people is greater than 10, we follow similar steps to those used in the previous problem but focus on finding the probability that the sample mean exceeds 10.

We will continue with the same assumptions and values:
- Population mean (\( \mu \)) = 12
- Population standard deviation (\( \sigma \)) = 4
- Sample size (\( n \)) = 25
- Sample mean (\( \bar{x} \)) = 10

1. **Calculate the standard error (SE)**:
   \[
   SE = \frac{\sigma}{\sqrt{n}} = \frac{4}{\sqrt{25}} = 0.8
   \]

2. **Calculate the z-score** using the formula:
   \[
   z = \frac{\bar{x} - \mu}{SE} = \frac{10 - 12}{0.8} = \frac{-2}{0.8} = -2.5
   \]

3. **Find the cumulative probability** corresponding to a z-score of -2.5:
   The cumulative probability for \( z = -2.5 \) is approximately 0.0062. This is the probability that the sample mean is less than 10.

4. **Find the probability that the sample mean is greater than 10**:
   Since the total probability for any event is 1, we subtract the cumulative probability from 1:
   \[
   P(\bar{x} > 10) = 1 - P(\bar{x} < 10) = 1 - 0.0062 = 0.9938
   \]

So, the probability that the mean of a sample of 25 people is greater than 10 is approximately 0.9938 or 99.38%.

Would you like more details or have any questions?

Here are 8 related questions you might find helpful:

1. How do we interpret a high probability result in hypothesis testing?
2. What is the role of the standard deviation in determining probabilities?
3. How does the sample size affect the shape of the sampling distribution?
4. What is the difference between cumulative probability and point probability?
5. How can we use the Central Limit Theorem for non-normal populations?
6. What are the common pitfalls in calculating probabilities using the normal distribution?
7. How does changing the population mean affect the probability calculations?
8. In what scenarios would we use a one-tailed test versus a two-tailed test?

**Tip:** Always check the assumptions behind your calculations, especially the distribution and sample size, to ensure the accuracy and validity of your probability results.

Learn how to calculate the probability that the mean of a sample of 25 people is greater than 10. Includes step-by-step solution using concepts like standard error, z-score, and cumulative probability. Suitable for advanced high school students or undergraduates.

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