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

Answer:

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.

Step-by-step explanation:

The proportion of the sample is

$p=\frac{114}{200}=0.57$

The standard deviation of the sample proportion is

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

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.