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Let's go through the two parts of the question.

### Part (a): Find the probability that a single randomly selected value is less than 147.9, given that the population has a normal distribution with mean \(\mu = 119.7\) and standard deviation \(\sigma = 81\).

We can solve this using the **Z-score** formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
where:
- \(X = 147.9\),
- \(\mu = 119.7\),
- \(\sigma = 81\).

Substitute the values:
\[
Z = \frac{147.9 - 119.7}{81}
\]

Now, we can calculate this Z-score and find the corresponding probability from the standard normal distribution table.

### Part (b): Find the probability that a randomly selected sample of size \(n = 40\) has a mean less than 147.9.

For sample means, the standard deviation changes according to the **standard error** of the mean:
\[
\sigma_{\text{mean}} = \frac{\sigma}{\sqrt{n}}
\]
Here:
- \(\sigma = 81\),
- \(n = 40\).

We calculate the standard error:
\[
\sigma_{\text{mean}} = \frac{81}{\sqrt{40}}
\]

Then, use the same Z-score formula but with the sample mean and standard error:
\[
Z = \frac{147.9 - 119.7}{\sigma_{\text{mean}}}
\]

After calculating this Z-score, we can also use the standard normal distribution to find the probability.

Let me now calculate the values.### Solutions:

- **Part (a):** The probability that a single randomly selected value is less than 147.9 is approximately \( P(X < 147.9) = 0.6361 \).
  
- **Part (b):** The probability that a randomly selected sample of size \(n = 40\) has a mean less than 147.9 is approximately \( P(M < 147.9) = 0.9862 \).

Let me know if you need further details or clarifications.

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Here are 5 related questions for practice:
1. How do you calculate the probability for a value greater than a given number in a normal distribution?
2. What happens to the standard error as the sample size increases?
3. Can you explain the difference between population standard deviation and sample standard deviation?
4. How do you interpret the Z-score value in context?
5. How would the probabilities change if the standard deviation was larger?

**Tip:** Remember that larger sample sizes reduce the standard error, leading to tighter confidence intervals for the sample mean.

A population of values has a normal distribution with μ = 119.7 and σ = 81. Find the probability that a single randomly selected value is less than 147.9. Also, find the probability that a randomly selected sample of size n = 40 has a mean less than 147.9.

Learn how to calculate probabilities for a normal distribution with mean 119.7 and standard deviation 81. This problem involves finding the probability that a value or sample mean is less than 147.9, using Z-scores and standard errors. Suitable for high school students in Grades 10-12.

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