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

To find the probability that the mean of a sample of 25 people is less than 10, we need to make some assumptions and follow these steps:

1. **Assume the population mean (μ) and standard deviation (σ)**: For a normally distributed population, we need the population mean and standard deviation. If these are not given, we can use hypothetical values or a specific distribution provided in a problem.

2. **Central Limit Theorem (CLT)**: The CLT states that the distribution of the sample mean will be normally distributed if the sample size is sufficiently large (typically n > 30 is considered, but n = 25 can be acceptable in some contexts), regardless of the population distribution, given the population has a finite variance. The mean of the sample means will be μ (population mean), and the standard deviation of the sample means (standard error) will be \( \frac{\sigma}{\sqrt{n}} \).

3. **Calculate the z-score**: This is the number of standard deviations the sample mean is away from the population mean. The formula for the z-score is:
   \[
   z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}
   \]
   where \( \bar{x} \) is the sample mean, \( \mu \) is the population mean, \( \sigma \) is the population standard deviation, and \( n \) is the sample size.

4. **Find the probability using the z-score**: Use the standard normal distribution table (z-table) or a calculator to find the probability corresponding to the calculated z-score.

Let's denote:
- Population mean: \( \mu \)
- Population standard deviation: \( \sigma \)
- Sample size: \( n = 25 \)
- Sample mean: \( \bar{x} = 10 \)

Assume hypothetical values for illustration:
- \( \mu = 12 \)
- \( \sigma = 4 \)

Now, calculate the standard error (SE):
\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{4}{\sqrt{25}} = \frac{4}{5} = 0.8
\]

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

Using the z-table or a standard normal distribution calculator, find the probability that corresponds to a z-score of -2.5.

The cumulative probability for \( z = -2.5 \) is approximately 0.0062.

So, the probability that the mean of a sample of 25 people is less than 10 is about 0.0062 or 0.62%.

Would you like more details or have any questions?

Here are 8 related questions you might find helpful:

1. How does the Central Limit Theorem apply to different sample sizes?
2. What is the importance of the standard error in statistical analysis?
3. How do you interpret a z-score in a normal distribution?
4. What assumptions must be met for the Central Limit Theorem to hold?
5. How can you use a z-table to find probabilities for different z-scores?
6. What is the difference between a sample mean and a population mean?
7. How does increasing the sample size affect the standard error and probability calculation?
8. What are the implications of a low probability result in hypothesis testing?

**Tip:** When dealing with probabilities and normal distributions, always ensure the assumptions of normality and the applicability of the Central Limit Theorem are met for accurate results.

Learn how to calculate the probability that the mean of a sample of 25 people is less than 10 using statistical methods. Understand the application of the Central Limit Theorem and z-score calculations with an example. Suitable for college-level students and those interested in probability theory and statistical analysis.

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