Question

Using a significance level of 0.05, test for a difference in the proportion of number of eligible citizens who registered to vote.
group        Number registered         sample size
no reminders                    70                      100
reminders                         80                       100

Answer 1

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p1 = p2
Alternate Hypothesis, Ha: p1 ≠ p2

Rejection Region
This is two tailed test, for α = 0.05
Critical value of z are -1.96 and 1.96.
Hence reject H0 if z < -1.96 or z > 1.96

p1cap = X1/N1 = 70/100 = 0.7
p1cap = X2/N2 = 80/100 = 0.8
pcap = (X1 + X2)/(N1 + N2) = (70+80)/(100+100) = 0.75

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.7-0.8)/sqrt(0.75*(1-0.75)*(1/100 + 1/100))
z = -1.63

P-value Approach
P-value = 0.1025
As P-value >= 0.05, fail to reject null hypothesis.