To solve these problems, we'll need to use the properties of the normal distribution and the concept of the z-score. However, you're missing key pieces of information:

1. The mean ($\mu$) of the population.
2. The standard deviation ($\sigma$) of the population.
3. The sample size ($n$).

Let me outline how to solve these problems:

Part 1: Probability for a Single Randomly Selected Value $P(X < 158.7)$

We can find this probability using the z-score formula for a single value:

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

Where:

• $X = 158.7$,
• $\mu = \text{population mean}$,
• $\sigma = \text{population standard deviation}$.

Once we calculate the z-score, we can look up the cumulative probability from the standard normal distribution table or use a calculator to find $P(X < 158.7)$.

Part 2: Probability for a Sample Mean $P(M < 158.7)$

For a sample of size $n$, the distribution of the sample mean $M$ is normally distributed with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt{n}}$ (called the standard error).

We calculate the z-score for the sample mean using:

$z = \frac{M - \mu}{\frac{\sigma}{\sqrt{n}}}$

Where:

• $M = 158.7$,
• $\mu = \text{population mean}$,
• $\sigma = \text{population standard deviation}$,
• $n = \text{sample size}$.

Then, just like in part 1, we can find the cumulative probability for this z-score.

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If you provide the values of $\mu$, $\sigma$, and $n$, I can compute the exact probabilities for you. Would you like to proceed with that?

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Here are some related questions to further your understanding:

1. How does the sample size affect the standard error of the mean?
2. What is the difference between $P(X < 158.7)$ and $P(M < 158.7)$?
3. Why do we use the standard normal distribution for finding probabilities?
4. How would the probability change if the population had a different standard deviation?
5. Can you explain why the sample mean has a smaller variability than individual values?

Tip: The larger the sample size, the smaller the standard error, which makes the sample mean distribution more concentrated around the population mean.