Question

Determining Sample Size The sample size needed to estimate the difference between two population proportions to within a margin of error E with a confidence level of 1 - a can be found by using the following expression:

\({\bf{E = }}{{\bf{z}}_{\frac{{\bf{\alpha }}}{{\bf{2}}}}}\sqrt {\frac{{{{\bf{p}}_{\bf{1}}}{{\bf{q}}_{\bf{1}}}}}{{{{\bf{n}}_{\bf{1}}}}}{\bf{ + }}\frac{{{{\bf{p}}_{\bf{2}}}{{\bf{q}}_{\bf{2}}}}}{{{{\bf{n}}_{\bf{2}}}}}} \)

Replace \({{\bf{n}}_{\bf{1}}}\;{\bf{and}}\;{{\bf{n}}_{\bf{2}}}\) by n in the preceding formula (assuming that both samples have the same size) and replace each of \({{\bf{p}}_{\bf{1}}}{\bf{,}}{{\bf{q}}_{\bf{1}}}{\bf{,}}{{\bf{p}}_{\bf{2}}}\;{\bf{and}}\;{{\bf{q}}_{\bf{2}}}\)by 0.5 (because their values are not known). Solving for n results in this expression:

\({\bf{n = }}\frac{{{\bf{z}}_{\frac{{\bf{\alpha }}}{{\bf{2}}}}^{\bf{2}}}}{{{\bf{2}}{{\bf{E}}^{\bf{2}}}}}\)

Use this expression to find the size of each sample if you want to estimate the difference between the proportions of men and women who own smartphones. Assume that you want 95% confidence that your error is no more than 0.03.

Short answer

The sample size for men and women is 2135.

Step-by-step solution

Given information

The formula for the sample size is given as,

\(n = \frac{{z_{\frac{\alpha }{2}}^2}}{{2{E^2}}}\)

Where, E represents margin of error and\({z_{\frac{\alpha }{2}}}\)is the critical value (two-tailed).

The margin of error is no more than 0.03 and the confidence level is 95% or 0.95.

Compute the critical value

The critical value\({z_{\frac{\alpha }{2}}}\)is defined at\(\alpha \)level of significance as,

\(P\left( {Z > {z_{\frac{\alpha }{2}}}} \right) = \frac{\alpha }{2}\)

As the confidence level is 0.95, the significance level is 0.05.

Thus, the critical value is,

\(\begin{array}{c}P\left( {Z > {z_{\frac{{0.05}}{2}}}} \right) = \frac{{0.05}}{2}\\P\left( {Z > {z_{\frac{{0.05}}{2}}}} \right) = 0.025\\1 - P\left( {Z < {z_{0.025}}} \right) = 0.025\\P\left( {Z < {z_{0.025}}} \right) = 0.975\end{array}\)

From the standard normal table, the critical value is hence obtained at the intersection of row 1.9 and column 0.06 which gives the z-score of 1.96.

Compute the sample size

Substitute the values in the given formula,

\(\begin{array}{c}n = \frac{{{{1.96}^2}}}{{2{{\left( {0.03} \right)}^2}}}\\ = 2134.22\\ \approx 2135\end{array}\)

Thus, the required sample size for men and women is 2135.