Expand Your Knowledge: Sample Size, Difference of Proportions What about the sample size n for confidence intervals for the difference of proportions p1 2p2? Let us make the following assumptions: equal sample sizes n 5n1 5n2 and all four quantities ˆ n1p1,

ˆ n1q1, ˆ n2 p2, and ˆ n2q2 are greater than 5. Those readers familiar with algebra can use the procedure outlined in Problem 31 to show that if we have preliminary estimates ˆp1 and ˆp2 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 ˆp1 and ˆp2, then the theory similar to that used in this section tells us that the sample size n should be at least

(a) In Problem 20 (Myers–Briggs personality type indicators in common for married couples), suppose we want to be 99% confident that our estimate ˆ ˆ p1 2p2 for the difference p1 2p2 has a maximal margin of error E 5 0.04. Use the preliminary estimates pˆ1 5 289/375 for the proportion of couples sharing two personality traits and pˆ2 5 23/571 for the proportion having no traits in common. How large should the sample size be (assuming equal sample size—i.e., n 5n1 5n2)?

(b) Suppose that in Problem 20 we have no preliminary estimates for ˆp1 and ˆp2 and we want to be 95% confident that our estimate ˆ ˆ p1 2p2 for the difference p1 2p2 has a maximal margin of error E 5 0.05. How large should the sample size be (assuming equal sample size—i.e., n 5n1 5n2)?AppendixLO1

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Z.c n = ( + P292). E