A nutritionist wants to estimate the difference between the proportion of males and females who consume the USDA's recommended daily intake of calcium. What sample size should be obtained if she wishes the estimate to be within 3 percentage points with \(95 \%\) confidence, assuming that

(a) she uses the results of the USDA's 1994-1996 Diet and Health Knowledge Survey, according to which \(51.1 \%\) of males and \(75.2 \%\) of females consume the USDA's recommended daily intake of calcium, and

(b) she does not use any prior estimates?

Approach We have \(E=0.03\) and \(z_{\frac{\alpha}{2}}=z_{0.05}^{2}=z_{0.025}=1.96\). To answer part (a), let \(\hat{p}_{1}=0.511\) (for males) and \(\hat{p}_{2}=0.752\) (for females) in Formula (4). To answer part (b), use Formula (5).

Formula 5

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= n = n = n2 = 0.5 == 0.5( E 2 (5)