To solve this, we need to break it down into two separate parts:

Part 1: Probability that a single randomly selected value $X$ is greater than 208.8

The problem provides us with the following information:

• The population has a normal distribution, with a given mean $\mu$ and standard deviation $\sigma$.
• We need to find $P(X > 208.8)$, which is the probability that a randomly selected value is greater than 208.8.

Unfortunately, you did not provide the mean $\mu$ and standard deviation $\sigma$ of the population. I’ll assume they are missing from the problem statement you copied, and you should provide these values.

The procedure for finding $P(X > 208.8)$:

1. Standardize the value to a $Z$-score using the formula:

   $Z = \frac{X - \mu}{\sigma}$

2. Use the standard normal distribution to find the probability associated with this $Z$-value.

3. Subtract the cumulative probability from 1, since you want $P(X > 208.8)$.

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Part 2: Probability that a sample mean $M$ is greater than 208.8

For the second part of the question, where you are asked to find the probability of the sample mean being greater than 208.8, you need the sample size $n$. Again, the value for $n$ is missing from the information provided.

In this case, we work with the sampling distribution of the sample mean, which has a mean $\mu$ and a standard deviation known as the standard error of the mean, calculated as:

$\text{Standard Error (SE)} = \frac{\sigma}{\sqrt{n}}$

The steps to find $P(M > 208.8)$:

1. Standardize the sample mean to a $Z$-score:

   $Z = \frac{M - \mu}{\text{SE}} = \frac{208.8 - \mu}{\frac{\sigma}{\sqrt{n}}}$

2. Use the standard normal distribution to find the corresponding probability.

3. As before, subtract the cumulative probability from 1 to find $P(M > 208.8)$.

To complete these calculations, please provide the following:

• The population mean $\mu$
• The population standard deviation $\sigma$
• The sample size $n$

Would you like me to proceed with any assumptions, or can you provide the missing values?

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Here are 5 related questions to expand understanding:

1. How do you calculate the standard error of the mean when the population standard deviation is known?
2. What is the difference between the population mean and sample mean?
3. How does the sample size affect the standard error of the mean?
4. Why do we subtract from 1 to find $P(X > \text{value})$ in a normal distribution?
5. How does the Central Limit Theorem justify using a normal distribution for sample means?

Tip: Always check if the problem is asking for probabilities of individual values or sample means; the approaches differ!