Preliminary analyses indicate that you can consider the assumptions for using nonpooled​ t-procedures satisfied. Researchers obtained the following data on the number of acute postoperative days in the hospital using the dynamic and static systems. Obtain a 90% confidence interval for the​ difference, mu 1 minus mu 2μ1−μ2​, between the mean numbers of acute postoperative days in the hospital with the dynamic and static systems.​ (Note: x overbar 1x1equals=7.42, s 1s1=1.88​, x overbar 2x2equals=11.00, and s 2s2=3.58)
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Part 1
The 90% confidence interval is from enter your response here to enter your response here.
​(Round to three decimal places as​ needed.)

To construct a 90% confidence interval for the difference in means between two independent groups, we will use the formula for the confidence interval in a two-sample \( t \)-test:

\[
CI = (\bar{x}_1 - \bar{x}_2) \pm t^* \cdot SE
\]

Where:
- \( \bar{x}_1 \) and \( \bar{x}_2 \) are the sample means for the two groups.
- \( t^* \) is the critical \( t \)-value for the specified confidence level and degrees of freedom.
- \( SE \) is the standard error of the difference in means.

The formula for the standard error of the difference is:

\[
SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}
\]

### Step-by-Step Solution:

1. **Given Data**:
   - \( \bar{x}_1 = 7.42 \), \( s_1 = 1.88 \), \( n_1 = 12 \)
   - \( \bar{x}_2 = 11.00 \), \( s_2 = 3.58 \), \( n_2 = 6 \)
   - Confidence level = 90%, so \( t^* \) will correspond to \( \alpha/2 = 0.05 \) for the degrees of freedom.

2. **Calculate the standard error (\( SE \))**:
   \[
   SE = \sqrt{\frac{1.88^2}{12} + \frac{3.58^2}{6}}
   \]

3. **Determine degrees of freedom** using the Welch–Satterthwaite equation:
   \[
   df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}}
   \]

4. **Find the critical \( t^* \)-value** for the calculated degrees of freedom.

5. **Compute the confidence interval** using:
   \[
   CI = (\bar{x}_1 - \bar{x}_2) \pm t^* \cdot SE
   \]

Let me calculate this step-by-step.import math
from scipy.stats import t

# Given data
x1 = 7.42
s1 = 1.88
n1 = 12
x2 = 11.00
s2 = 3.58
n2 = 6
confidence_level = 0.90

# Calculate standard error (SE)
SE = math.sqrt((s1**2 / n1) + (s2**2 / n2))

# Calculate degrees of freedom using Welch–Satterthwaite equation
numerator = (s1**2 / n1 + s2**2 / n2)**2
denominator = ((s1**2 / n1)**2 / (n1 - 1)) + ((s2**2 / n2)**2 / (n2 - 1))
df = numerator / denominator

# Critical t-value for the confidence level
alpha = 1 - confidence_level
t_star = t.ppf(1 - alpha / 2, df)

# Confidence interval
mean_diff = x1 - x2
margin_of_error = t_star * SE
CI_lower = mean_diff - margin_of_error
CI_upper = mean_diff + margin_of_error

CI_lower, CI_upper, SE, df(-6.574485076898247,
 -0.5855149231017536,
 1.5590381650235507,
 6.4184522879071775)(-6.574485076898247,
 -0.5855149231017536,
 1.5590381650235507,
 6.4184522879071775)successThe 90% confidence interval for the difference in means (\( \mu_1 - \mu_2 \)) is:

\[
(-6.574, -0.586)
\]

### Additional Details:
- **Standard Error (SE):** \( 1.559 \)
- **Degrees of Freedom (df):** \( 6.418 \)

If you have further questions or need clarification, let me know!

### 5 Related Questions:
1. How does changing the confidence level (e.g., to 95%) affect the confidence interval?
2. What assumptions are required for using a nonpooled \( t \)-procedure?
3. Why is the Welch–Satterthwaite equation used for degrees of freedom in this case?
4. How would increasing the sample sizes affect the confidence interval?
5. Can this method be used if the variances between groups are equal?

### Tip:
When constructing confidence intervals, always verify the underlying assumptions (e.g., normality and independence) for more accurate results.

Preliminary analyses indicate that you can consider the assumptions for using nonpooled t-procedures satisfied. Researchers obtained the following data on the number of acute postoperative days in the hospital using the dynamic and static systems. 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. (Note: x̄1 = 7.42, s1 = 1.88, x̄2 = 11.00, and s2 = 3.58).

Learn how to calculate a 90% confidence interval for the difference in mean postoperative days in hospital using dynamic and static systems. Includes step-by-step calculations with formulas for standard error, degrees of freedom, and critical t-value. Key data points: x̄1 = 7.42, s1 = 1.88, n1 = 12, x̄2 = 11.00, s2 = 3.58, n2 = 6.

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