Hi, i was struggling with this question. Could you please help me how to solve i

Question

Hi, i was struggling with this question. Could you please help me how to solve it?

Ancient Athens has three social classes who were included in holding political offices. Members of the Zeugitae, one of the three classes, are upset because they believe they were under-represented in last year's Boule compared to the prior year (the Boule is a body of 500 citizens chosen randomly among those eligible to hold office and who together administer the day-to-day public matters of Athens). To settle the dispute with the other tribes, the Zeugitae agree to randomly select members who served on the two Boulai in question and compare the results to see if there was a significant difference in their level of representation. For the year being argued over, 50 of the surveyed members reported being Zeugitae, or 0.25 of those asked. For the prior year, 100 of the surveyed members reported being Zeugitae, or 0.33 of those asked. Just that instant, you, the time-travelling social scientist, arrive to share the wonders of modern statistics with the Ancient Athenians. You do the math to a 0.95-level of certainty. Was there a significant difference in the Zeugitae's representation between the two Boulai? How do you know?

Explanation / Answer

Given that,
sample one, x1 =50, n1 =200, p1= x1/n1=0.25
sample two, x2 =100, n2 =303, p2= x2/n2=0.33
null, Ho: p1 = p2
alternate, H1: p1 != p2
level of significance, = 0.05
from standard normal table, two tailed z /2 =1.96
since our test is two-tailed
reject Ho, if zo < -1.96 OR if zo > 1.96
we use test statistic (z) = (p1-p2)/(p^q^(1/n1+1/n2))
zo =(0.25-0.33)/sqrt((0.298*0.702(1/200+1/303))
zo =-1.92
| zo | =1.92
critical value
the value of |z | at los 0.05% is 1.96
we got |zo| =1.92 & | z | =1.96
make decision
hence value of |zo | < | z | and here we do not reject Ho
p-value: two tailed ( double the one tail ) - Ha : ( p != -1.9202 ) = 0.0548
hence value of p0.05 < 0.0548,here we do not reject Ho
ANSWERS
---------------
null, Ho: p1 = p2
alternate, H1: p1 != p2
test statistic: -1.92
critical value: -1.96 , 1.96
decision: do not reject Ho
p-value: 0.0548

Conclusion:
there was no significant difference in the Zeugitae's representation between the two Boulai.

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