Question

In the 2015 federal election, 39.5% of the electorate voted for the Liberal party, 31.9% for the Conservative party, 19.7% for the NDP, 4.7% for the Bloc Quebecois and 3.5% for the Green party. The most recent pool as of the launch of the 2019 election campaign shows a tie between the Liberals and the Conservatives at 33.8%. This pool was based on 1185 respondents.

(a) Based on this recent pool, test whether this is sufficient evidence to conclude that the level of support for the conservatives has increased since the last election. Use the 5% level of significance and show your manual calculations.

(b) Using recent pool data, build an appropriate 95% one-sided confidence interval for the true proportion of support for the conservatives. Is this CI consistent with your conclusion in a) above?

(c) Would your conclusion be the same as in a) above if you had used a 10% confidence level for the hypothesis test?

(d) Now, suppose you want to estimate the national level of support for the Liberals at the start of the 2019 campaign using a 95% 2-sided confidence interval with a margin of error of  1% based on the results of the last election, what sample size would be required

(e) Would the sample size calculated above be sufficient to estimate the support for the Bloc Quebecois within the same level of confidence and margin of error? If not, how many more respondents would you need?

Answer 1

In the 2015 federal election, 39.5% of the electorate voted for the Liberal party, 31.9% for the Conservative party, 19.7% for the NDP, 4.7% for the Bloc Quebecois and 3.5% for the Green party. The most recent pool as of the launch of the 2019 election campaign shows a tie between the Liberals and the Conservatives at 33.8%. This pool was based on 1185 respondents.

(a) Based on this recent pool, test whether this is sufficient evidence to conclude that the level of support for the conservatives has increased since the last election. Use the 5% level of significance and show your manual calculations.

Ho: P=0.319, H₁: P> 0.319

p=0.338

=1.40

Table value of z at 0.05 level = 1.645

Rejection Region: Reject Ho if z > 1.645

Calculated z =1.40   , not in the rejection region

The null hypothesis is not rejected.

There is not sufficient evidence to conclude that the level of support for the conservatives has increased since the last election.

(b) Using recent pool data, build an appropriate 95% one-sided confidence interval for the true proportion of support for the conservatives. Is this CI consistent with your conclusion in a) above?

CI for proportion

P =0.338

Z value for 95% = 1.96

=( 0.3111, 0.3649)

95% confidence interval =(0.3111, 0.3649)

(c) Would your conclusion be the same as in a) above if you had used a 10% confidence level for the hypothesis test?

No.

Table value of z at 0.10 level = 1.282

Rejection Region: Reject Ho if z > 1.282

Calculated z =1.40 falls in the rejection region

The null hypothesis is rejected

(d) Now, suppose you want to estimate the national level of support for the Liberals at the start of the 2019 campaign using a 95% 2-sided confidence interval with a margin of error of ± 1% based on the results of the last election, what sample size would be required

p=0.338

For 95%, z=1.96

d=0.01

Sample size = (z²*p*(1-p))/d²

= (1.96²*0.338*0.662)/0.01²

=8595.81

The sample size required= 8596

(e) Would the sample size calculated above be sufficient to estimate the support for the Bloc Quebecois within the same level of confidence and margin of error? If not, how many more respondents would you need?

p=0.047

For 95%, z=1.96

d=0.01

Sample size = (z²*p*(1-p))/d²

= (1.96²*0.047*0.953)/0.01²

=1720.69

The sample size required= 1721