Question

Use the expression in the accompanying discussion of sample size to find the size of each sample if you want to estimate the difference between proportions of men and women who own smartphones. Assume that you want 99% confidence that your error is no more than 0.025. Click the icon to view the discussion of sample size. The sample should include (Type whole numbers.) men and women.

Answer 1

Solution

$\small \mathbf{given:}\;Margin\;of\;error\;(E)=0.025$

$\mathbf{for \;99\% \;confidence \;level, \;z_{\frac{\alpha }{2}}=2.576}$

a=1-C=1-0.99 = 0.01 $\frac{\alpha}{2} =\frac{0.01}{2} =\mathbf{0.005}$ Refer Standard normal table/Z-table or use excel function "=NORM.S.INV((1-0.005))" to find the Z-value. $\mathbf{z_{\frac{\alpha }{2}}=2.576}$

$\small \mathbf{formula:}\;Sample\;size\;\left (n \right )=\frac{z_{\frac{\alpha }{2}}^{2}}{2E^{2}}$

$\small n=\frac{(2.576)^{2}}{2*(0.025)^{2}}$

$\small n=\frac{6.635776}{2*0.000625}$

$\small n=5308.6208$

$\small \mathbf{n\approx {\color{Red} 5309}}$

$\small \mathbf{The \:sample\: should\: include \:{\color{Red} 5309 }\:men \:and \:{\color{Red} 5309 }\:women}$