What about sample size? If we want a confi­dence interval with maximal margin of error E and level of confi­dence c, then Section 7.1 shows us which formulas to apply for a single mean m and Section 7.3 shows us formulas for a single proportion p.

(a) How about a difference of means? When s₁ and s₂ are known, the margin of error E for a c% con­fidence interval is

Let us make the simplifying assumption that we have equal sample sizes n so that n = n₁ = n₂. We also assume that n ³ 30. In this context, we get

Solve this equation for n and show that

(b) In Problem 15 (football and basketball player heights), suppose we want to be 95% sure that our estimate x̅₁ – x̅₂ for the difference m₁ – m₂ has a margin of error E – 0.05 foot. How large should the sample size be (assuming equal sample size— i.e., n = n₁ = n₂)? Since we do not know s₁ or s₂ and n ³ 30, use s₁ and s₂, respectively, from the preliminary sample of Problem 15.

(c) In Problem 16 (petal lengths of two iris species), suppose we want to be 90% sure that our estimate x̅₁ – x̅₂ for the difference m₁ – m₂ has a margin of error E = 0.1 cm. How large should the sample size be (assuming equal sample size—i.e., n – n₁ – n₂)? Since we do not know s₁ or s₂ and n ³ 30, use s₁ and s₂, respectively, from the preliminary sample of Problem 16.

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