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### Scenario 1: **Income of Retail Managers**

1. **What is the best point estimate of the population mean?**
   - The best point estimate of the population mean is the **sample mean**.
   - In this case, the sample mean annual income of the retail managers is **$45,420**. Therefore, the best estimate of the population mean is **$45,420**.

2. **What is a reasonable range of values for the population mean?**
   - To find a reasonable range of values, we need to construct a **95% confidence interval** for the population mean.
   
   Given:
   - Sample mean ($\bar{x}$) = 45,420
   - Population standard deviation ($\sigma$) = 2,050
   - Sample size ($n$) = 49

   The formula for the confidence interval is:
   \[
   CI = \bar{x} \pm Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}
   \]
   Where $Z_{\alpha/2} = 1.96$ for a 95% confidence level.
   
   Let's calculate the margin of error:
   \[
   MOE = 1.96 \times \frac{2,050}{\sqrt{49}} = 1.96 \times \frac{2,050}{7} = 1.96 \times 292.86 = 573.01
   \]

   Now, construct the confidence interval:
   \[
   45,420 \pm 573.01 \quad \Rightarrow \quad (44,846.99, 45,993.01)
   \]

   Therefore, a reasonable range for the population mean is **$44,847 to $45,993**.

3. **What do these results mean?**
   - These results indicate that we are **95% confident** that the true population mean of annual incomes of retail managers lies between **$44,847 and $45,993**. This range provides a reasonable estimate for the population mean based on the sample data.

---

### Scenario 2: **Tire Tread Life**

1. **Construct a 95 percent confidence interval for the population mean.**
   - For the tire manufacturer, we need to construct a 95% confidence interval for the population mean tread remaining.
   
   Given:
   - Sample mean ($\bar{x}$) = 0.32 inch
   - Standard deviation ($\sigma$) = 0.09 inch
   - Sample size ($n$) = 10

   The confidence interval formula is:
   \[
   CI = \bar{x} \pm Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}
   \]
   Where $Z_{\alpha/2} = 1.96$ for a 95% confidence level.
   
   Let's calculate the margin of error:
   \[
   MOE = 1.96 \times \frac{0.09}{\sqrt{10}} = 1.96 \times \frac{0.09}{3.162} = 1.96 \times 0.02846 = 0.0558
   \]

   Now, construct the confidence interval:
   \[
   0.32 \pm 0.0558 \quad \Rightarrow \quad (0.2642, 0.3758)
   \]

   Therefore, the 95% confidence interval for the population mean tread remaining is approximately **(0.264 to 0.376 inches)**.

2. **Would it be reasonable for the manufacturer to conclude that after 50,000 miles the population mean amount of tread remaining is 0.30 inches?**
   - Yes, it would be reasonable to conclude that the population mean tread remaining is **0.30 inches** after 50,000 miles because **0.30 inches falls within the 95% confidence interval** of **(0.264 to 0.376 inches)**. Therefore, it is plausible that the true population mean could be 0.30 inches.

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Would you like more detailed calculations or further explanations? Here are 5 questions to explore:

1. How does increasing the confidence level (e.g., to 99%) affect the confidence interval?
2. What assumptions are required when using a confidence interval with a known standard deviation?
3. How does increasing the sample size impact the width of the confidence interval?
4. What would happen if the standard deviation was unknown for the population?
5. How can manufacturers use confidence intervals to make more informed decisions?

### Tip:
When constructing a confidence interval, remember that a **larger sample size** reduces the margin of error, resulting in a **narrower confidence interval** and a more precise estimate.

Confidence interval for a mean, σ known - What is the best point estimate of the population mean? What is a reasonable range of values for the population mean? What do these results mean? Construct a 95 percent confidence interval for the population mean. Would it be reasonable for the manufacturer to conclude that after 50,000 miles the population mean amount of tread remaining is 0.30 inches?

Explore how to construct a 95% confidence interval for a population mean using real-world examples. Learn how to estimate the mean annual income of retail managers and the remaining tire tread life after 50,000 miles. Discover key statistical concepts, including margin of error, sample mean, and population standard deviation.

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