# Are 96% of Americans really on Facebook?

A recent study by Bank of America found that the proportion of Americans under 50 on Facebook is 96%. However, the second paragraph of the article questions the validity of the study. It says:

A total of 418 participants were included in the polling. Such a limited sample size will surely bolster calls that the survey was statistically unreliable, but the persons questioned were supposed to “roughly resemble” US demographics as a whole.

But is a sample size of 418 really limited? Are the statistics really unreliable? We can use Minitab Statistical Software to find out! I entered the summarized data into a 1 Proportion test to get a 95% confidence interval for the true proportion of Americans under 50 that are on Facebook:

Here are the results:

When we look at the confidence interval, we can be 95% confident that the true proportion of Americans under 50 that are on Facebook is between 93.6% and 97.6%. That doesn’t sound unreliable to me!

Now say you work for a business that will advertise on Facebook as long as at least 90% of Americans under 50 are on the social network. Again, we can use Minitab to do a hypothesis test on this data.

And we get the output below.  Using this data, what decision would you make?

We see that the p-value is less than our significance level of 0.05. So again, the sample size is plenty big enough to conclude that the true proportion of Americans under 50 that are on Facebook is greater than 90%.

Now, keep in mind all of these statistics assume that the 418 participants were randomly chosen from the population of all Americans under 50. Otherwise the results are not valid. For example,  if the survey was done online, then Americans that didn’t have connection to the Internet would have been left out of the study. This sample would be biased and wouldn’t be representative of the population.

The article says only that the persons questioned were supposed to “roughly resemble” US demographics as a whole. Without knowledge of how the study was actually conducted, we are unable to determine if the sample was truly random. So although there is still a chance the statistics are unreliable, we’ve been able to show that it isn’t because of the sample size!

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