\(\displaystyle{p}_{{1}}\): Proportion of families who emigrated to the U.S. from Eastern Europe five years ago

\(\displaystyle{p}_{{2}}\): Proportion of families who emigrated to the U.S. from Pacific islands five years ago

Null hypothesis: \(\displaystyle{H}_{{0}}:{p}_{{1}}={p}_{{2}}\)

Sample 1: emigrated to the U.S. from Eastern Europe five years ago

Number families who emigrated to the U.S. from Eastern Europe five years ago: \(\displaystyle{n}_{{1}}={235}\)

Number families who now own homes: \(\displaystyle{x}_{{1}}={165}\)

Proportion of families who emigrated to the U.S. from Eastern Europe five years ago :small \(\displaystyle{w}{i}{d}{e}\hat{{{p}}}_{{{1}}}={0.7021}\)

Sample 2: emigrated to the U.S. from Pacific islands five years ago

Number families who emigrated to the U.S. from Pacific islands five years ago: \(\displaystyle{n}_{{2}}={195}\)

Number families who now own homes: \(\displaystyle{x}_{{2}}={125}\)

Sample proportion of families who emigrated to the U.S. from Pacific islands five years ago: small \(\displaystyle{w}{i}{d}{e}\hat{{{p}}}_{{{2}}}=\frac{{125}}{{195}}={0.641}\)

Test Statistic: \(\displaystyle{Z}={\frac{{\hat{{p}}_{{1}}\ -\ \hat{{p}}_{{2}}}}{{\sqrt{{{\frac{{\hat{{p}}_{{1}}\ {\left({1}-\hat{{p}}_{{1}}\right)}}}{{{n}_{{1}}}}}\ +\ {\frac{{\hat{{p}}_{{2}}\ {\left({1}-\hat{{p}}_{{2}}\right)}}}{{{n}_{{2}}}}}}}}}}\)

Test Statistic: \(\displaystyle{Z}={\frac{{{0.7021}\ -\ {0.641}}}{{\sqrt{{{\frac{{{0.7021}{\left({1}-{0.7021}\right)}}}{{{235}}}}\ +\ {\frac{{{0.641}{\left({1}-{0.641}\right)}}}{{{195}}}}}}}}}\ =\ {\frac{{{0.0611}}}{{{0.0455}}}}\ =\ {1.3429}\)

For two failed test:

\(\displaystyle{P}-{V}{a}{l}{u}{e}\ =\ {P}{\left({Z}\ {<}\ -{Z}_{{{s}{t}{a}{t}}}\right)}\ +\ {P}{\left({Z}\ {>}\ {Z}_{{{s}{t}{a}{t}}}\right)}\)

\(\displaystyle={2}\times\ {P}{\left({Z}{<}-{1.3429}\right)}={2}\times\ {0.0897}={0.1794}\)

p-value \(\displaystyle=\ {0.1794}\)

\(\displaystyle\alpha\ =\ {0.05}\)

As P-Value i.e. is greater than Level of significance i.e \(\displaystyle{\left({P}-{v}{a}{l}{u}{e}:{0.1794}{>}{0.01}\right.}\):Level of significance), Fail to Reject Null Hypothesis

There is not sufficient evidence to conclude that there is a significant difference in home ownership rates in the two groups of immigrants.