What is the minimal sample size needed for a 95% confidence interval to have a maximal margin of error of 0.2 in the following scenarios? (a) a preliminary estimate for p is 0.37 (b) there is no preliminary estimate for p Step 1 (a) What is the minimal sample size needed for a 95% confidence interval to have a maximal margin of error of 0.2 if a preliminary estimate for p is 0.377 Recall that the minimal sample size n for a given maximal margin of error when a preliminary estimate for p is known can be found using the formulan p(1-p) zc E 2, where E is the maximal margin of error. We are interested in the minimal sample size for a 95% confidence interval with a maximal margin of error of 0.2. The table below lists critical values for confidence intervals at different levels of confidence. Confidence Interval Critical Values zc Level of Confidence c Critical Value zc 0.70, or 70% 1.04 0.75, or 75% 1.15 0.80, or 80% 1.28 0.85, or 85% 1.440.90, or 90% 1.645 0.95, or 95% 1.96 0.98, or 98% 2.33 0.99, or 99% 2.58 According to this table, for a 95% confidence level, z0.95 1.96. We want the confidence interval to have a maximal margin of error of 0.2 and the preliminary estimate for p is 0.37. Therefore, E and p. Apply these values to the formula for n. Remember that because we are looking for the minimal sample size, we must round the final answer up the nearest whole number. n = p(1-p) zc E 2 = 0.37(1-0.37) 1.962 =
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