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

(a) The null and alternative hypothesis

Test statistic

where  

Thus

= 1.41

At 5% level of significance , right tail critical value of z is 1.65 (from z table)

Since calculated z < critical value

We fail to reject H0.

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

b) 95% confidence interval for true proportion of supporters of conservatives

For 95% confidence interval , zc = 1.96

Therefore 95% confidence interval is

=

= (0.311, 0.365)

= (31.1% , 36.5%)

We can see that the confidence interval includes the original proportion of supporters ( 31.9%) , there is not sufficient evidence to conclude that support for conservatives has increased since last election.

This confidence interval is consistent with the conclusion in part (a)

(c) For 10% confidence level , zc =1.65

Therefore 95% confidence interval is

=

= (0.315, 0.361)

= (31.5% , 36.1%)

We can see that the confidence interval includes the original proportion of supporters ( 31.9%) , there is not sufficient evidence to conclude that support for conservatives has increased since last election.

This confidence interval is consistent with the conclusion in part (a)

(d) margin of error =

for 95% confidence , zc =1.96

Given , margin of error =1% or 0.01

Thus

(rounding)

Sample size =8596

(e)

margin of error should be 0.0.1

confidence 95%

We can see that margin of error is not equal to 0.01

Thus sample size required will be different

Calculation of sample size

(rounded)

We need 1298 respondents