Question

Surveys have been widely used by politicians around the world as a way of monitoring the opinions of the electorate. Six months ago, a survey was undertaken to determine the degree of support for a national party leader. Of a sample of 1,100, 56% indicated that they would vote for this politician. This month, another survey of 800 voters revealed that 46% now support the leader.

1. At the 5% significance level, can we infer that that the national leader’s popularity has decreased?
2. At the 5% significance level, can we infer that the national leader’s popularity has decreased by more than 3%?
3. Estimate with 90% confidence the decrease in percentage support between now and 6 months ago.

Answer 1

Ans:

pooled proportion=(0.56*1100+0.46*800)/(1100+800)=0.5179

a)

Test statistic:

z=(0.56-0.46)/sqrt(0.5179*(1-0.5179)*((1/1100)+(1/800)))

z=4.31

p-value=P(z>4.31)=0.0000

As,p-value<0.05,we reject the null hypothesis and can conclude that  the national leader’s popularity has decreased.

b)

Test statistic:

z=((0.56-0.46)-0.03)/sqrt(0.5179*(1-0.5179)*((1/1100)+(1/800)))

z=3.01

p-value=P(z>3.01)=0.0013

As,p-value<0.05,we reject the null hypothesis and can conclude that the national leader’s popularity has decreased by more than 3%.

c)

90% confidence interval for difference in proportion

=(0.56-0.46)+/-1.645*sqrt((0.56*(1-0.56)/1100)+(0.46*(1-0.46)/800))

=0.100+/-0.038

=(0.062, 0.138)