Question

A random sample of 200 voters in a town is selected, and 114 are found to support an annexa- tion suit. Find the 96% confidence interval for the fraction of the voting population favoring the suit.

(b) What can we assert with 96% confidence about the possible size of our error if we estimate the fraction of voters favoring the annexation suit to be 0.57?

Answer 1

a) 96% CI:
`0.51\leq\pi\leq 0.63`

b) If we estimate that the fraction of voters is 0.57, we can claim with 96% confidence that the error is equal or less than 0.06 from the estimated proportion.

Explanation:

The proportion of the sample is

`p=(114)/(200)=0.57`

The standard deviation of the sample proportion is

`\sigma=\sqrt{(p(1-p))/(n) } =\sqrt{(0.57(1-0.57))/(200) } =\sqrt{(0.2451)/(200) } =0.035`

For a 96% CI, the z-value is z=1.751.

Then, the 96% CI can be written as:

`p-z\cdot \sigma\leq\pi\leq p+z\cdot \sigma\\\\0.57-1.751*0.035\leq\pi\leq 0.57+1.751*0.035\\\\0.57-0.06\leq\pi\leq 0.57+0.06\\\\0.51\leq\pi\leq 0.63`

b) If we estimate that the fraction of voters is 0.57, we can claim with 96% confidence that the error is equal or less than 0.06 from the estimated proportion.