Two candidates A and Bare running for elections in a very large district. No pol

Question

Two candidates A and Bare running for elections in a very large district. No polls have been conducted before. You want to test the claim that candidate A is going to win the election by choosing a significance level alpha = 0.01. Thus you ask 500 randomly selected likely voters to secretely cast their ballots the day before the election. a) If you find 255 votes for candidate A and 245 votes for candidate B are you going to accept the claim? b) If you find 270 votes for candidate A and 230 votes for candidate B are you going to accept the claim?

Explanation / Answer

1.
Given that,
sample one, x1 =245, n1 =500, p1= x1/n1=0.49
sample two, x2 =255, n2 =500, p2= x2/n2=0.51
null, Ho: p1 = p2
alternate, H1: p1 != p2
level of significance, a = 0.01
from standard normal table, two tailed z a/2 =
since our test is two-tailed
reject Ho, if zo < -2.576 OR if zo > 2.576
we use test statistic (z) = (p1-p2)/v(p^q^(1/n1+1/n2))
zo =(0.49-0.51)/sqrt((0.5*0.5(1/500+1/500))
zo =-0.632
| zo | =0.632
critical value
the value of |z a| at los 0.01% is 2.576
we got |zo| =0.632 & | z a | =2.576
make decision
hence value of |zo | < | z a | and here we do not reject Ho
p-value: two tailed ( double the one tail ) - Ha : ( p != -0.6325 ) = 0.5271
hence value of p0.01 < 0.5271,here we do not reject Ho

ANSWERS
---------------
null, Ho: p1 = p2
alternate, H1: p1 != p2
test statistic: -0.632
critical value: -2.576 , 2.576
decision: do not reject Ho
p-value: 0.5271

Q2.
Given that,
sample one, x1 =270, n1 =500, p1= x1/n1=0.54
sample two, x2 =230, n2 =500, p2= x2/n2=0.46
null, Ho: p1 = p2
alternate, H1: p1 != p2
level of significance, a = 0.01
from standard normal table, two tailed z a/2 =
since our test is two-tailed
reject Ho, if zo < -2.576 OR if zo > 2.576
we use test statistic (z) = (p1-p2)/v(p^q^(1/n1+1/n2))
zo =(0.54-0.46)/sqrt((0.5*0.5(1/500+1/500))
zo =2.53
| zo | =2.53
critical value
the value of |z a| at los 0.01% is 2.576
we got |zo| =2.53 & | z a | =2.576
make decision
hence value of |zo | < | z a | and here we do not reject Ho
p-value: two tailed ( double the one tail ) - Ha : ( p != 2.5298 ) = 0.0114
hence value of p0.01 < 0.0114,here we do not reject Ho

ANSWERS
---------------
null, Ho: p1 = p2
alternate, H1: p1 != p2
test statistic: 2.53
critical value: -2.576 , 2.576
decision: do not reject Ho
p-value: 0.0114

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