From the course: Everyday Statistics, with Eddie Davila

Proportions of coins

From the course: Everyday Statistics, with Eddie Davila

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Proportions of coins

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- [Instructor] In my office I have a change bin. When I do use cash, I accumulate change and it goes in the change bin. I throw everything in there. And in a pinch, I use some on a vending machine. Usually I grab the quarters and maybe some dimes. Based on random cash payments and the change I get, plus the occasional removal of coins, I wonder how many of the coins in the bin are quarters, or dimes, or any coin. I decided to use statistics to see if I could get some guidance on the matter. I took a small sample of coins, I just grabbed a small handful hoping this would qualify as a random sample. I happened to grab 15 coins. As you can see, I grabbed five quarters, two dimes, five nickels, and three pennies. I converted those numbers into proportions. I then decided to create 95% confidence intervals for my proportions. Here's the formula I used to create my confidence intervals, my upper and lower limits. For quarters, it would be the proportion of quarters, 0.20, plus or minus my z score for 95%, 1.96, times the sampling error, which is the proportion of quarters divided by the square root of the sample size, 15 coins. I did this for every type of coin in my sample. Here are my results. So this tells me that based on my sample of 15 coins, I'm 95% confident there are between 16.5% and 50.2% quarters in the container. And between 6.6% and 20.1% dimes. This seemed like a really small sample size, so before I counted the coins, I repeated this process with a bigger sample. This time I scooped up a much larger sample, 114 coins. My large sample had 10 quarters, 27 dimes, 20 nickels, 57 pennies. The associated proportions are also listed. I then computed the 95% confidence interval for each proportion of coins based on this larger sample of 114 coins. With these two very different samples and my two sets of 95% confidence intervals, I dumped my coins into a coin counting machine so I could compare the actual proportions to those of my calculated confidence intervals. Here were the actual number of coins in my container. Right away, I see something I didn't expect. There was one single dollar coin. That didn't show up in any of my samples. So I didn't even create a confidence interval for dollar coins. That outcome was completely unexpected. Now let's compare the actual data to the confidence intervals based on 15 coins. Not very good at all. The actual proportion of dollars, quarters, dimes, and pennies all fell outside of my calculated intervals. Only the proportion of nickels was within the limits. As I said, not very good. Perhaps the limits of my larger sample size confidence intervals fared better. These did fare better, but still, there is a concern. First, the proportion of dollar coins was outside the limit. That seems reasonable as that single coin was quite an anomaly. But the quarters again fell outside the limits. It was close, but still outside the upper limit. So what does this mean? Well perhaps our first sample was too small. And in reality, once the calculations for one coin are compromised, the other limits are impacted. How about the fact that even our larger sample didn't provide us expected results for quarters? Well first, these are 95% confidence intervals. The actual number fell outside the limits, but not by very much. The bigger issue may be the random sample. Perhaps grabbing a bunch of coins with your hand may not be the best method for taking a sample. Maybe small coins or large coins have a greater chance of staying in your hand when you just reach in and grab a handful. Developing confidence intervals that are reliable requires a few things. Understanding that even good 95% confidence intervals will fail 5% of the time. And perhaps more importantly, confidence intervals are reliant on quality data. If your random sample was small and if your random sample was perhaps not really random, then perhaps your confidence intervals should not be trusted. Next time you see a confidence interval, question the data collection. And remember, even a good 95% confidence interval has its limits.

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