Question

In a random sample of 1000 Canadians conducted by Ekos polling, 412 indicated that they would support a liberal party in the next federal election.

a) Construct a 90% confidence interval for the proportion of Canadians voting liberal.

b) Write a sentence explaining the meaning of the interval you constructed in part a.

c) How large a sample is needed for a sampling error of +/-4 percentage points with 90% confidence?

Answer 1

Given that, n = 1000 and x = 412

sample proportion $(\hat p)$ = 412/1000 = 0.412

a) A 90% confidence level has significance level of 0.10 and critical value is,

Za/2 = 1.645

The 90% confidence interval for p is,

d - 1 *d * U'Z + p = (1 -2) - - Za/2 + 1 n

0.412 + (1 -0.412) 10.412 + (1 -0.412) 0.412-1.645*1/ << p < 0.412+1.645*1/ 1000 1000

0.412 - 0.0256 <p < 0.412 + 0.0256

0.3864 <p < 0.4376

The 90% confidence interval is (0.3864, 0.4376)

b) We are 90% confident that the true proportion of all Canadians voting liberal is from 0.3864 to 0.4376.

c) margin of error (E) = 0.04

We want to find, the sample size (n),

n=p* (1-)* 2)2 = 0.412 + (1 -0.412) * 1.645 -)' = 409.7192 0.04

n410

Therefore, required sample size is 410