Question # Solve given Inferences involving two populations Inference about two Population Proportions: Independent Samples: A government housing agency is compa

Comparing two groups
ANSWERED Solve given Inferences involving two populations Inference about two Population Proportions:
Independent Samples:
A government housing agency is comparing home ownership rates among several immigrant groups. In a sample of 235 families who emigrated to the U.S. from Eastern Europe five years ago, 165 now own homes. In a sample of 195 families who emigrated to the U.S. from Pacific islands five years ago, 125 now own homes. 2020-12-06

$$\displaystyle{p}_{{1}}$$: 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.