The image contains a problem related to probability or statistics, based on the visible context. Here’s what the text on the screen says:

Question:
Note that when $n = 10$, the sample mean $\bar{x}$ may take the values $\bar{x}_1, \bar{x}_2, \dots, \bar{x}_k$. Find the value of $k$.

To answer this question, consider that the sample mean $\bar{x}$ is the average of 10 random variables, and therefore it can take a finite number of values based on the possible outcomes of these variables.

Options:

• 100
• 55
• 62
• 52
• 51

To solve this problem:

1. Understanding the problem: The sample mean $\bar{x}$ is the average of 10 independent and identically distributed (i.i.d.) random variables. The number of different values the sample mean can take depends on the number of different values that each variable can take.

2. Possible Values of Each Variable: If each variable can take a certain finite number of distinct values, then the mean will also have a specific number of distinct possible values.

3. Determine $k$: The number $k$ represents the number of distinct possible values for the sample mean $\bar{x}$. Without additional information about the distribution of the random variables, an exact calculation isn't straightforward. However, this type of problem typically involves combinatorial reasoning or considering the possible outcomes of the variables.

Since the problem does not provide specific values for the variables, it's likely that $k$ is found using reasoning based on possible averages of 10 values.

If you'd like further detailed explanation or have more details, please let me know! Here are some related questions that could deepen your understanding:

1. What does the Central Limit Theorem imply about the distribution of the sample mean $\bar{x}$ as $n$ increases?
2. How do you calculate the number of distinct possible sample means if each observation can take only a finite number of values?
3. In what situations can the number of distinct possible sample means $k$ be greater or less than the number of distinct observations?
4. What assumptions are made about the distribution of the random variables in calculating $k$?
5. How does increasing the sample size $n$ affect the number of distinct possible values of $\bar{x}$?

Tip: When working with sample means, always consider the properties of the distribution of the original data, as this influences the possible outcomes and their probabilities.