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.