Step 1

Given:

Number of women \(\displaystyle{\left({n}_{{{1}}}\right)}={172}\)

Number of men \(\displaystyle{\left({n}_{{{2}}}\right)}={45}\)

Number of women used coupons \(\displaystyle{\left({x}_{{{1}}}\right)}={74}\)

Number of men used coupons \(\displaystyle{\left({x}_{{{2}}}\right)}={12}\)

Confidence level \(\displaystyle={0.95}\)

So, \(\displaystyle\alpha={1}-{0.95}={0.05}\)

Step 2

Sample proportions

\(\displaystyle\hat{{{p}}}_{{{1}}}={\frac{{{x}_{{{1}}}}}{{{n}_{{{1}}}}}}={\frac{{{74}}}{{{172}}}}\)

\(\displaystyle\hat{{{p}}}_{{{2}}}={\frac{{{x}_{{{2}}}}}{{{n}_{{{2}}}}}}={\frac{{{12}}}{{{45}}}}\)

From z tables, \(\displaystyle{P}{\left({z}{>}{1.96}\right)}={0.025}\)

Step 3

Finding confidence interval

\(\displaystyle{\left(\hat{{{p}}}_{{{1}}}-\hat{{{p}}}_{{{2}}}\right)}-{z}_{{{\frac{{\alpha}}{{{2}}}}}}\times\sqrt{{{\frac{{\hat{{{p}}}_{{{1}}}\times{\left({1}-\hat{{{p}}}_{{{1}}}\right)}}}{{{n}_{{{1}}}}}}+{\frac{{\hat{{{p}}}_{{{2}}}\times{\left({1}-\hat{{{p}}}_{{{2}}}\right)}}}{{{n}_{{{2}}}}}}}},{\left(\hat{{{p}}}_{{{1}}}-\hat{{{p}}}_{{{2}}}\right)}+{z}_{{\frac{{\alpha}}{{{2}}}}}\times\sqrt{{{\frac{{\hat{{{p}}}_{{{1}}}\times{\left({1}-\hat{{{p}}}_{{{1}}}\right)}}}{{{n}_{{{1}}}}}}+{\frac{{\hat{{{p}}}_{{{2}}}\times{\left({1}-\hat{{{p}}}_{{{2}}}\right)}}}{{{n}_{{{2}}}}}}}}\)

\(\displaystyle{\left({\frac{{{74}}}{{{172}}}}-{\frac{{{12}}}{{{45}}}}\right)}-{1.96}\times\sqrt{{{\frac{{{\frac{{{74}}}{{{172}}}}\times{\left({1}-{\frac{{{74}}}{{{172}}}}\right)}}}{{{172}}}}+{\frac{{{\frac{{{12}}}{{{45}}}}\times{\left({1}-{\frac{{{12}}}{{{45}}}}\right)}}}{{{45}}}}}},{\left({\frac{{{74}}}{{{172}}}}-{\frac{{{12}}}{{{45}}}}\right)}+{1.96}\)

\(\displaystyle\times\sqrt{{{\frac{{{\frac{{{74}}}{{{172}}}}\times{\left({1}-{\frac{{{74}}}{{{172}}}}\right)}}}{{{172}}}}+{\frac{{{\frac{{{12}}}{{{45}}}}\times{\left({1}-{\frac{{{12}}}{{{45}}}}\right)}}}{{{45}}}}}}\)

0.0147, 0.3125

The confidence interval for difference in women and men proportions of using coupons is 0.0147, 0.3125.