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

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

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{(\pi(1-\pi))/(n)}`

In which

z is the zscore that has a pvalue of
`1 - (\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 = (10)/(100) = 0.1`

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)} = 0.1 - 1.96\sqrt{(0.1*0.9)/(100)} = 0.0412`

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