The problem seems to involve finding the probability of a sample mean lying within a certain range. However, the question is incomplete. Could you clarify what specific probability you're looking for? For example:

• Are you asking for the probability that the sample mean falls within a specific interval?
• Are you asking about the sampling distribution of the sample mean?
• Is there a specific value or range for which you want the probability calculated?

In the meantime, here's some helpful information:

Sampling Distribution of the Sample Mean:

1. Population Mean ($\mu$): 76
2. Population Variance ($\sigma^2$): 256
   Thus, $\sigma = \sqrt{256} = 16$
3. Sample Size ($n$): 100

The sample mean ($\bar{X}$) for a random sample of size $n$ from a population with mean $\mu$ and standard deviation $\sigma$ follows a normal distribution (Central Limit Theorem applies due to the large sample size): $\bar{X} \sim N\left(\mu, \frac{\sigma}{\sqrt{n}}\right)$

$\bar{X} \sim N\left(76, \frac{16}{\sqrt{100}}\right) = N(76, 1.6)$

Using this information, probabilities can be calculated based on the standard normal distribution.

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Let me know the range or specific probability you need, and I’ll calculate it! 😊

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Follow-up Questions:

1. What interval or range are you interested in for the sample mean?
2. Should we find a one-tailed or two-tailed probability?
3. Do you want to see the full process for converting to a $z$-score?
4. Would you like to explore the concept of the Central Limit Theorem in more detail?
5. Do you need help calculating similar probabilities for other scenarios?

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Tip: Always check whether the population is finite or infinite when applying statistical formulas. In this case, the assumption of an infinite population simplifies the calculations.