To calculate the 99.9% confidence interval for $\mu_1 - \mu_2$, we use the formula for the confidence interval for the difference between two means, assuming independent samples:

$CI = (\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2} \cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

Given Data:

• $n_1 = 50, \bar{x}_1 = 2.9, s_1 = 0.85$
• $n_2 = 55, \bar{x}_2 = 2.76, s_2 = 0.76$
• Confidence level: 99.9% $\rightarrow \alpha = 0.001$

Steps:

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

2. Find the critical value $t_{\alpha/2}$: Using the degrees of freedom and the confidence level.

3. Compute the margin of error (ME): $ME = t_{\alpha/2} \cdot \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

4. Calculate the confidence interval: $CI = (\bar{x}_1 - \bar{x}_2) \pm ME$

Let me compute this step-by-step.import scipy.stats as stats import numpy as np

Given data

n1, x1_bar, s1 = 50, 2.9, 0.85 n2, x2_bar, s2 = 55, 2.76, 0.76 alpha = 0.001 # For 99.9% confidence level

Calculate degrees of freedom using Welch-Satterthwaite equation

s1_sq_over_n1 = (s12) / n1 s2_sq_over_n2 = (s22) / n2

df = ((s1_sq_over_n1 + s2_sq_over_n2)2) / ( (s1_sq_over_n12) / (n1 - 1) + (s2_sq_over_n2**2) / (n2 - 1) )

Critical t-value

t_critical = stats.t.ppf(1 - alpha / 2, df)

Margin of error

margin_of_error = t_critical * np.sqrt(s1_sq_over_n1 + s2_sq_over_n2)

Confidence interval

difference_means = x1_bar - x2_bar ci_lower = difference_means - margin_of_error ci_upper = difference_means + margin_of_error

ci_lower, ci_upper(-0.3957694546343645, 0.6757694546343648)(-0.3957694546343645, 0.6757694546343648)successThe 99.9% confidence interval for $\mu_1 - \mu_2$ is:

$(-0.40, 0.68)$

Interpretation:

We are 99.9% confident that the true difference between the two population means ($\mu_1 - \mu_2$) lies between $-0.40$ and $0.68$.

Do you have any questions or need further clarifications? Here are some related questions to expand on:

1. How do we interpret the confidence interval when it includes zero?
2. What assumptions are made in this calculation?
3. How does changing the confidence level affect the interval?
4. How is the Welch–Satterthwaite formula for degrees of freedom derived?
5. Can this method be applied when sample sizes are very different?

Tip: Increasing the sample size reduces the margin of error, resulting in a narrower confidence interval.