Question

In a sample of 400 voters, 360 indicated they favor the incumbent governor. The 95% confidence interval of voters not favoring the incumbent is 0.871 to 0.929
0.120 to 0.280

0.765 to 0.835

0.071 to 0.129

Answer 1

Answer:

0.071 to 0.129

Step-by-step explanation:

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

$\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}$

In which

z is the zscore that has a pvalue of $1 - \frac{\alpha}{2}$.

For this problem, we have that:

In a sample of 400 voters, 360 indicated they favor the incumbent governor. This means that 400-360 = 40 do not favor the incumbent governor. So $n = 400, \pi = \frac{40}{400} = 0.1$

95% confidence interval

So $\alpha = 0.05$, z is the value of Z that has a pvalue of $1 - \frac{0.05}{2} = 0.975$, so $Z = 1.96$.

The lower limit of this interval is:

$\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.10 - 1.96\sqrt{\frac{0.10*0.90}{400}} = 0.071$

The upper limit of this interval is:

$\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.10 + 1.96\sqrt{\frac{0.10*0.90}{400}}= 0.129$

The correct answer is:

0.071 to 0.129