To find the endpoints of the interval for the expression $\mu_1 - \mu_2$, we need to use the given context to establish the lower and upper bounds.

However, there seems to be missing information, specifically the values or ranges for $\mu_1$ and $\mu_2$. In general, intervals for differences between means (like $\mu_1 - \mu_2$) are calculated using specific data or statistical measures such as confidence intervals.

Given the prompt, I'll assume the task involves calculating the confidence interval for the difference between two population means, $\mu_1$ and $\mu_2$. If additional statistical data were provided, we could compute it as follows:

1. Determine the sample means $\bar{x}_1$ and $\bar{x}_2$.
2. Calculate the standard errors for each sample mean.
3. Use the formula for the confidence interval for the difference between two means.

Since specific values are missing, I will guide you through the typical steps and formula used:

General Formula

For a confidence interval of the difference between two means, the formula is typically:

$(\bar{x}_1 - \bar{x}_2) \pm z \cdot \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$

Where:

• $\bar{x}_1$ and $\bar{x}_2$ are the sample means.
• $\sigma_1^2$ and $\sigma_2^2$ are the variances of the two populations.
• $n_1$ and $n_2$ are the sample sizes.
• $z$ is the z-score corresponding to the desired confidence level (for example, 1.96 for 95%).

Example

Without concrete numbers, I'll use an example. Let's assume:

• $\bar{x}_1 = 50$
• $\bar{x}_2 = 45$
• $\sigma_1 = 5$
• $\sigma_2 = 5$
• $n_1 = 100$
• $n_2 = 100$
• Confidence level = 95% (z = 1.96)

Step-by-Step Calculation:

1. Calculate the difference in sample means: $\bar{x}_1 - \bar{x}_2 = 50 - 45 = 5$

2. Calculate the standard error: $SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} = \sqrt{\frac{5^2}{100} + \frac{5^2}{100}} = \sqrt{\frac{25}{100} + \frac{25}{100}} = \sqrt{0.25 + 0.25} = \sqrt{0.5} \approx 0.707$

3. Calculate the margin of error: $ME = z \cdot SE = 1.96 \cdot 0.707 \approx 1.385$

4. Calculate the confidence interval: $(\bar{x}_1 - \bar{x}_2) \pm ME = 5 \pm 1.385$ $\text{Lower bound} = 5 - 1.385 \approx 3.615$ $\text{Upper bound} = 5 + 1.385 \approx 6.385$

Rounded to the Nearest Integer:

$\text{Lower bound} \approx 4$ $\text{Upper bound} \approx 6$

So, the interval is approximately $(4, 6)$.

If you have specific values for $\mu_1$ and $\mu_2$, or other statistical data, please provide them to refine this calculation.

Do you have any specific values or data to include, or should we proceed with the assumed example?

Follow-up Questions:

1. What are the sample means $\bar{x}_1$ and $\bar{x}_2$?
2. What are the standard deviations $\sigma_1$ and $\sigma_2$?
3. What are the sample sizes $n_1$ and $n_2$?
4. What is the confidence level you are working with?
5. Is there any specific statistical test you are referring to?
6. Are there any assumptions about the population distributions?
7. Are you using data from a specific study or dataset?
8. Do you need the exact formula or a step-by-step guide for your specific data?

Tip:

Always verify the assumptions of your statistical tests, such as normality or equal variances, before calculating confidence intervals to ensure the results are valid.