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The image shows a problem related to the construction of a 90% confidence interval for the difference between the means of two independent samples (dynamic and static systems) in a nonpooled t-procedure.

### Given Information:
- Sample means:
  - \( \bar{x}_1 = 7.00 \) (Dynamic system)
  - \( \bar{x}_2 = 10.83 \) (Static system)
- Sample standard deviations:
  - \( s_1 = 1.71 \)
  - \( s_2 = 3.97 \)
- Sample sizes are not given directly, but we can deduce them from the table:
  - \( n_1 = 8 \) (Dynamic system)
  - \( n_2 = 9 \) (Static system)

### Formula for the Confidence Interval:
The formula for the confidence interval for the difference between two independent means when using the nonpooled t-procedure is:

\[
(\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2} \times \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}
\]

Where:
- \( t_{\alpha/2} \) is the critical value from the t-distribution, with degrees of freedom approximated using:

\[
df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1-1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2-1}}
\]

Let's calculate the confidence interval step by step.The 90% confidence interval for the difference between the mean number of acute postoperative days in the hospital (dynamic system - static system) is approximately:

\[
(-6.440, -1.220)
\]

This means we are 90% confident that the true difference between the means lies between -6.440 and -1.220.

The degrees of freedom used for this calculation is approximately 11.13.

Would you like more details or have any further questions?

Here are 5 related questions for further exploration:
1. What does a negative confidence interval indicate about the difference between the two systems?
2. How would the confidence interval change if the sample sizes were larger?
3. How is the nonpooled t-procedure different from a pooled t-procedure?
4. What would the confidence interval look like for a 95% confidence level?
5. How does the margin of error relate to the width of the confidence interval?

**Tip:** Always check if the assumptions of the t-procedure (like normality and independence of samples) are met when interpreting results.

Obtain a 90% confidence interval for the difference, μ1 - μ2, between the mean numbers of acute postoperative days in the hospital with the dynamic and static systems.

Learn how to calculate a 90% confidence interval for the difference between two means using a nonpooled t-procedure. The problem compares the mean postoperative days between dynamic and static systems, providing a step-by-step solution with degrees of freedom approximation.

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