So, Probably, Maybe
In late 2011 and into 2012, a wave started in Canada, worked its way south to the States and then throughout the world. The wave was Carly Rae Jepsen’s hit “Call Me Maybe”. In the hit, Carly sings for her crush to call her, maybe. Carly Rae Jepsen played at the revamped Mall of America on August 22nd, which has revived the Call Me Maybe fever in the Twin Cities. Many people have wondered if Carly was wasting her time waiting for her crush to call. Well, as it would turn out, we can estimate with a 95% confidence level the percentage range within the population of “crushees” that do indeed call their “crushers” based on a sample.
In order to start, we need a sample. We are going to invoke the Central Limit Theorem, which tells us that we will have normal or near normal distribution of a sampling, if the sample is large enough. We will also make the assumption that the overall population distribution would be generally bell shaped as opposed to being badly skewed, multiple peaks, so that a non-bias sampling of 120 crushees is adequately sufficient to get a proper and representative distribution sampling of the true population distribution. Out of our 120 crushees 46 of them (38.33%) said that they would indeed call their respective crusher. So, we have a sample and we have a proportion result. Now, let’s get our terminology and perform our calculations. The sample size of 120 will be “n” and we will use the percentage representation of the 46 that would call, which is .3833, and let that value be referred to as “p-bar”.
Now, that we have n and p-bar, the last value for our formula is the value of “z”. This is where our friend Excel really shines. The “z” value we need for our proportion formula is the inverse of the standard normal cumulative distribution and we get it using these steps:
- Take 1-the confidence level (.95) in this case which equals 0.05
- Divide the result by two (for two sides of a bell curve) and this equals 0.025
- Use the excel function NORMSINV(1-.025) and this gives us our “z” value of 1.96
Now we have everything we need to predict to a 95% confidence level that the true population of crushees that call their crushers falls within a certain range.
The formula that will give us our calculation is: (p-bar) plus or minus (z)(square root of (p-bar(1-pbar)/n)).
- The first step is to calculate ((p-bar(1-pbar))/n). In excel terms this would be =(((0.3833)*(1-0.3833))/120), which equals 0.00197
- We then square root it by using =SQRT(0.00197), which equals 0.044383
- Then we multiply that value by z by using =(0.044383)*(1.96). Which is 0.086991.
- This is our plus or minus value that gives us our upper and lower range. We add and subtract this to our p-bar value (0.3833) for our range calculation
The result is that we can say with 95% confidence that the actual number of crushees that call back their crusher will fall between the range 29.63% and 47.02%. Roughly, one third to one half are not bad odds that Carly may indeed get a call. Peace of mind brought to you by statistics.