Question

You recently sent out a survey to determine if the percentage of adults who use social media has changed from 66%, which was the percentage of adults who used social media five years ago. Of the 2809 people who responded to survey, 1634 stated that they currently use social media. Use the data from this survey to construct a 98% confidence interval estimate of the proportion of adults who use social media.

Answer 1

The 98% confidence interval estimate of the proportion of adults who use social media is (0.56, 0.6034).

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)`.

Of the 2809 people who responded to survey, 1634 stated that they currently use social media.

This means that
`n = 2809, \pi = (1634)/(2809) = 0.5817`

98% confidence level

So
`\alpha = 0.02`, z is the value of Z that has a pvalue of
`1 - (0.02)/(2) = 0.99`, so
`Z = 2.327`.

The lower limit of this interval is:

`\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.5817 - 2.327\sqrt{(0.5817*4183)/(2809)} = 0.56`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.5817 + 2.327\sqrt{(0.5817*4183)/(2809)} = 0.6034`

The 98% confidence interval estimate of the proportion of adults who use social media is (0.56, 0.6034).