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This problem involves calculating probabilities using a normal distribution and statistical software. Let's break it down step by step.

### Key Information:
- **Population Mean (μ)** = 15.1 weeks
- **Population Standard Deviation (σ)** = 5.1 weeks
- **Sample Size (n)** = 60 (for the second part)

The questions are:
1. **Find the probability that a single randomly selected value is between 15 and 17.1 weeks**.
2. **Find the probability that the sample mean for a sample size of 60 is between 15 and 17.1 weeks**.

### Part 1: Probability for a Single Value
This part involves calculating the probability for a **single value** from a normal distribution. We can standardize the range \(15 < X < 17.1\) using the z-score formula:

\[
z = \frac{X - \mu}{\sigma}
\]

#### Calculate Z-Scores:
1. For \(X = 15\):
   \[
   z_1 = \frac{15 - 15.1}{5.1} = -0.0196
   \]
2. For \(X = 17.1\):
   \[
   z_2 = \frac{17.1 - 15.1}{5.1} = 0.3922
   \]

Now, use statistical technology (like a calculator or software) to find the probability corresponding to these z-scores, i.e., the probability that \(z_1 < Z < z_2\).

### Part 2: Probability for the Sample Mean
For a sample of size 60, the **standard error (SE)** of the mean is calculated as:

\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{5.1}{\sqrt{60}} = 0.6584
\]

Now, we standardize the range \(15 < \bar{X} < 17.1\) using the z-score formula for the sample mean:

\[
z = \frac{\bar{X} - \mu}{SE}
\]

#### Calculate Z-Scores:
1. For \(\bar{X} = 15\):
   \[
   z_1 = \frac{15 - 15.1}{0.6584} = -0.1519
   \]
2. For \(\bar{X} = 17.1\):
   \[
   z_2 = \frac{17.1 - 15.1}{0.6584} = 3.0379
   \]

Again, use statistical software to find the probability corresponding to these z-scores, i.e., \(P(z_1 < Z < z_2)\).

### Next Steps:
You can use tools like Excel, a graphing calculator, or online calculators to compute the exact probabilities. Would you like assistance with finding the exact values for these probabilities or any clarifications?

### Related Questions:
1. How do you interpret z-scores in relation to probabilities?
2. How does increasing the sample size affect the standard error?
3. What would happen to the probability if the standard deviation were higher?
4. Can we use the z-distribution for any sample size?
5. What assumptions are necessary for using the normal distribution here?

### Tip:
Always check if you're dealing with a single value or a sample mean. The formulas for standard deviation and standard error differ accordingly!

A leading magazine reported that the average number of weeks an individual is unemployed is 15.1 weeks, with a standard deviation of 5.1 weeks. What is the probability that a single randomly selected value and a sample of size 60 is between 15 and 17.1 weeks?

Calculate the probability that a single randomly selected value or the mean of a sample of 60 unemployed individuals falls between 15 and 17.1 weeks using the normal distribution, Z-scores, and the concept of standard error. Detailed step-by-step guide with statistical software.

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