What about the sample size n for confi­dence intervals for the difference of proportions p₁ – p₂? Let us make the following assumptions: equal sample sizes n = n₁ = n₂ and all four quantities n₁p₁, n₁q₁, n₂p₂, and n₂q₂ are greater than 5. Those readers familiar with algebra can use the procedure outlined in Problem 28 to show that if we have preliminary estimates ¹ and ² and a given maximal margin of error E for a specifi­ed con­fidence level c, then the sample size n should be at least

However, if we have no preliminary estimates for ₁ and ₂, then the theory similar to that used in this section tells us that the sample size n should be at least

(a) In Problem 17 (Myers–Briggs personality type indicators in common for married couples), suppose we want to be 99% confi­dent that our estimate ₁ – ₂ for the difference p₁ – p₂ has a maximal margin of error E = 0.04. Use the preliminary estimates ₁ = 289/375 for the proportion of couples sharing two personality traits and ₂ = 23/571 for the proportion having no traits in common. How large should the sample size be (assuming equal sample size—i.e., n – n₁ – n₂)?

(b) Suppose that in Problem 17 we have no preliminary estimates for ₁ and ₂ and we want to be 95% con­fident that our estimate ₁ – ­₂ for the difference p₁ – p₂ has a maximal margin of error E = 0.05. How large should the sample size be (assuming equal sample size—i.e., n – n₁ – n₂)?

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