Question

We are formulating a 95% confidence interval for the population proportion. We have taken a random sample of 100 observations and observed 10 instances of the characteristic in question. What would be the lower interval limit for our 95% confidence interval? (NOTE: Round your estimated standard error to 3 decimal places. Round your final answer to 4 decimal places)

Answer 1

Answer:

The lower interval limit for our 95% confidence interval is of 0.0412.

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 level of $1-\alpha$, 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 $1 - \frac{\alpha}{2}$.

We have taken a random sample of 100 observations and observed 10 instances of the characteristic in question.

This means that $n = 100, \pi = \frac{10}{100} = 0.1$

95% confidence level

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:

$\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.1 - 1.96\sqrt{\frac{0.1*0.9}{100}} = 0.0412$

The lower interval limit for our 95% confidence interval is of 0.0412.