from math import sqrt from scipy.stats import t

Given data

n1 = 35 x1_bar = 2.07 s1 = 0.37

n2 = 25 x2_bar = 2.01 s2 = 0.44

Confidence level

confidence_level = 0.998 alpha = 1 - confidence_level

Degrees of freedom using Welch-Satterthwaite equation

df = ((s12 / n1 + s22 / n2)2) / ( (s12 / n1)2 / (n1 - 1) + (s22 / n2)**2 / (n2 - 1) )

Critical t value

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

Difference in means

mean_diff = x1_bar - x2_bar

Standard error of the difference

se_diff = sqrt(s12 / n1 + s22 / n2)

Confidence interval

margin_of_error = t_critical * se_diff lower_bound = mean_diff - margin_of_error upper_bound = mean_diff + margin_of_error

lower_bound_rounded = round(lower_bound, 2) upper_bound_rounded = round(upper_bound, 2)

(lower_bound_rounded, upper_bound_rounded)(-0.29, 0.41)(-0.29, 0.41)successThe 99.8% confidence interval for $\mu_1 - \mu_2$ is:

$-0.29 \pm 0.41$

This indicates the range $-0.29$ to $0.41$.

Would you like further clarification on this calculation? Here are some related questions:

1. What does the 99.8% confidence interval mean in practical terms?
2. How is the t-critical value determined for different confidence levels?
3. Why is the Welch-Satterthwaite equation used for degrees of freedom?
4. How do sample sizes affect the width of a confidence interval?
5. Can this method be applied to non-normal data distributions?

Tip: Always ensure the assumptions for using t-distributions are met, such as independence and approximate normality of the sample data.