I have two groups, with 100 people in one group(1) and 80 people in the other group(2). 42% of group 1 love cake. I am wondering what proportion of the 80 group 2 members would need to love cake for me to have 80% power to reject the null hypothesis that both of these groups have the same proportion of cake lovers (I hope I worded that right).
I did a power analysis in g-power (z test of proportions, proportion 1=42%, alphpa=.05, power=80%), and it suggested that about 63% of people in group 2 would need to be cake lovers for me to have 80% power to have statistically significant findings.
I was curious and wanted to see how many participants I would need to do a chi-squares test. I needed an effect size so I imputed my proposed proportions (41%, 63%), which appears to be a w value of about .43 (effect size). When I put this in the power analysis in g-power (chi-square goodness of fit), it indicated that I would need 70 people to have 80% power to detect a statistically significant difference of .43 magnitude.
This confused me. On the one hand I have 180 people and apparently I need proportions resembling 42%/63% to have 80% power. On the other hand, these proportions seem to reflect a w effect size of .43, which requires only 52 people to have 80% power.
What explains this difference? Does it have to do with the fact that on one hand I am trying to determine a proportion, and on the other hand I am trying to determine required sample size? These seem to be different questions, but I can't wrap my head around why the findings differ.