Question

A survey was given to a random sample of 1350 residents of a town to determine whether they support a new plan to raise taxes in order to increase education spending. Of those surveyed, 26% of the people said they were in favor of the plan. Determine a 95% confidence interval for the percentage of people who favor the tax plan, rounding values to the nearest tenth.

Answer 1

`0.26 - 1.96\sqrt{(0.26(1-0.26))/(1350)}=0.237`

`0.26 + 1.96\sqrt{(0.26(1-0.26))/(1350)}=0.283`

Rounded the 95% confidence interval would be given by (0.2;0.3)

Explanation:

We know that the estimated proportion of people said they were in favor of the plan is
`\hat p=0.26`

We want to construct a confidence interval with 95% of confidence, our significance level would be given by
`\alpha=1-0.95=0.05` and
`\alpha/2 =0.025`. And the critical value would be given by:

`z_(\alpha/2)=-1.96, z_(1-\alpha/2)=1.96`

The confidence interval for the true proportion of interest is given by the following formula:

`\hat p \pm z_(\alpha/2)\sqrt{(\hat p (1-\hat p))/(n)}`

Replacing the info given we got:

`0.26 - 1.96\sqrt{(0.26(1-0.26))/(1350)}=0.237`

`0.26 + 1.96\sqrt{(0.26(1-0.26))/(1350)}=0.283`

Rounded the 95% confidence interval would be given by (0.2;0.3)

Answer 2

The 95% confidence interval for the percentage of people who favor the tax plan is (23.7%, 28.3%).

Explanation:

Confidence interval for the proportion:

In a sample with a number n of people surveyed with a probability of a success of
`\pi`, and a confidence level of
`1-\alpha`, we have the following confidence interval of proportions.

`\pi \pm z\sqrt{(\pi(1-\pi))/(n)}`

In which

z is the zscore that has a pvalue of
`1 - (\alpha)/(2)`.

For this problem, we have that:

`n = 1350, \pi = 0.26`

95% confidence level

So
`\alpha = 0.05`, z is the value of Z that has a pvalue of
`1 - (0.05)/(2) = 0.975`, so
`Z = 1.96`.

The lower limit of this interval is:

`\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.26 - 1.96\sqrt{(0.26*0.74)/(1350)} = 0.237`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.26 + 1.96\sqrt{(0.26*0.74)/(1350)} = 0.283`

For the percentage:

Multiply the proportions by 100%.

0.237*100 = 23.7%.

0.283*100 = 28.3%.

The 95% confidence interval for the percentage of people who favor the tax plan is (23.7%, 28.3%).