Question

As part of the Pew Internet and American Life Project, researchers conducted two surveys in late 2009. The first survey asked a random sample of 800 U.S. teens about their use of social media and the internet. A second survey posed similar questions to a random sample of 2253 U.S. adults. In these two studies, 73% of teens and 47% of adults said that they use social networking sites. Use these results to construct and interpret a 95% confidence interval for the difference between the proportion of all U.S. teens and adults who use social networking sites.

Answer 1

The 95% confidence interval for the difference between the proportion of all U.S. teens and adults who use social networking sites is (0.223, 0.297). This means that we are 95% sure that the true difference of the proportion is in this interval, between 0.223 and 0.297.

Explanation:

Before building the confidence interval we need to understand the central limit theorem and the subtraction of normal variables.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean
`\mu = p` and standard deviation
`s = \sqrt{(p(1-p))/(n)}`

Subtraction between normal variables:

When two normal variables are subtracted, the mean is the difference of the means, while the standard deviation is the square root of the sum of the variances.

Sample of 800 teens. 73% said that they use social networking sites.

This means that:

`p_T = 0.73, s_T = \sqrt{(0.73*0.27)/(800)} = 0.0157`

Sample of 2253 adults. 47% said that they use social networking sites.

This means that:

`p_A = 0.47,s_A = \sqrt{(0.47*0.53)/(2253)} = 0.0105`

Distribution of the difference:

`p = p_T - p_A = 0.73 - 0.47 = 0.26`

`s = √(s_T^2+s_A^2) = √(0.0157^2+0.0105^2) = 0.019`

Confidence interval:

Is given by:

`p \pm zs`

In which

z is the zscore that has a pvalue of
`1 - (\alpha)/(2)`.

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`.

Lower bound:

`p - 1.96s = 0.26 - 1.96*0.019 = 0.223`

Upper bound:

`p + 1.96s = 0.26 + 1.96*0.019 = 0.297`

The 95% confidence interval for the difference between the proportion of all U.S. teens and adults who use social networking sites is (0.223, 0.297). This means that we are 95% sure that the true difference of the proportion is in this interval, between 0.223 and 0.297.