Question

Researchers ask a random sample of 1,001 adults nationwide whether they favor or oppose the legalization of marijuana. Fifty-five of respondents say they oppose it. If the researchers increase the sample size of the poll by a factor of 4 (to n=4,004), which of the following effects on the length of the 95% confidence interval for the proportion is most likely to be observed?
a. It will decrease by a factor of 4
b. It will decrease by a factor of 2
c. It will increase by a factor of 2
d. It will increase by a factor of 4

Answer 1

Answer:

b. It will decrease by a factor of 2

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 $1 - \frac{\alpha}{2}$.

The lower end of the interval is given by:

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

The upper end of the interval is given by:

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

The length of the interval is the subtraction of the upper end by the lower end, so it is:

$L = 2z\sqrt{\frac{\pi(1-\pi)}{n}}$

This means that the length is inverse proportional to the square root of the size of the sample.

So, if the sample size is multiplied by 4, the length of the interval is going to decrease by a factor of 2.