Consider two independent normal distributions. A random sample of size n₁ = 20 from the –first distribution showed x₁ = 12 and a random sample of size n₂ = 25 from the second distribution showed x₂ = 14.

(a) Check Requirements If s₁ and s₂ are known, what distribution does m₁ – m₂ follow? Explain.

(b) Given s₁ = 3 and s₂ = 4, fi­nd a 90% confidence interval for m¹ – m².

(c) Check Requirements Suppose s¹ and s² are both unknown, but from the random samples, you know s₁ = 3 and s₂ = 4. What distribution approximates the x̅₁ ­– x̅₂ distribution? What are the degrees of freedom? Explain.

(d) With s₁ = 3 and s₂ = 4, fi­nd a 90% con­fidence interval for m₁m₂.

(e) If you have an appropriate calculator or computer software, fi­nd a 90% con­fidence interval for m₁ – m₂ using degrees of freedom based on Satterthwaite’s approximation.

(f) Interpretation Based on the con­fidence intervals you computed, can you be 90% confident that m₁ is smaller than m₂? Explain