A population of values has a normal distribution with 
 and 
. You intend to draw a random sample of size 
.

Find the probability that a single randomly selected value is less than 158.7.
P(X < 158.7) = 


Find the probability that a sample of size 
 is randomly selected with a mean less than 158.7.
P(M < 158.7) = 


Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

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.

---

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.

A population of values has a normal distribution with mean (μ) and standard deviation (σ). You intend to draw a random sample of size (n). Find the probability that a single randomly selected value is less than 158.7. P(X < 158.7) = ? Find the probability that a sample of size n is randomly selected with a mean less than 158.7. P(M < 158.7) = ? Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Learn how to calculate the probability that a single randomly selected value is less than 158.7 and that a sample mean is less than 158.7 using the normal distribution. This problem involves key statistical concepts such as the Z-score, normal distribution, and the central limit theorem. Suitable for college-level students studying probability and statistics.

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