From the course: Everyday Statistics, with Eddie Davila
Proportions of coins
From the course: Everyday Statistics, with Eddie Davila
Proportions of coins
- [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.
Contents
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Political polls3m 14s
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Different sports, different stats4m 42s
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Test scores2m 59s
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Data collection2m 40s
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Birthdays in the USA3m 15s
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The house always wins4m 1s
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Wisdom of the crowd3m 12s
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The pay gap at Uber3m 58s
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Cancer survival rates4m 22s
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Television ratings4m 15s
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Historic stats stories4m 28s
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The one percent3m 58s
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New Year's Eve3m 28s
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Influenza3m 17s
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Winter is coming3m 11s
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The Super Bowl4m 39s
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Genetics3m 45s
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Relationships3m 48s
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The box office2m 39s
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Unemployment2m 41s
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Waiting in lines4m 29s
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Sleep2m 56s
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March Madness4m 43s
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Measuring what's important in business3m 28s
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Baseball4m 41s
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Income tax statistics1m 49s
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College waiting lists4m 26s
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The normal distribution is everywhere2m 53s
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Horse racing statistics3m 45s
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Statistics and the insurance industry2m 35s
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Restaurant statistics2m 43s
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Catching criminals with statistics2m 55s
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Getting directions from statistics2m 43s
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Lyme disease2m 55s
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Quality control in manufacturing2m 36s
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Zoo animal statistics2m 41s
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Earthquakes2m 54s
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Statistics of hunting2m 48s
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Transcontinental convoy2m 7s
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Moon landing3m 44s
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Summer movies2m 56s
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Reliability3m 47s
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Woodstock3m 44s
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Hurricanes2m 47s
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P-hacking2m 49s
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Salaries2m 51s
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Dow Jones3m
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Population3m 23s
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Epidemiology2m 55s
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Rock stars3m 44s
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Acceptance sampling3m 18s
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The value of my change4m 42s
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In-game win probabilities3m 39s
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Stock market ups and downs3m 18s
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Prohibition3m 30s
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Bayes' theorem4m 15s
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Divorce3m 35s
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The U.S. Census3m 26s
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English3m 8s
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Santa Claus3m 47s
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Parenting3m 6s
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Proportions of coins4m 53s
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Safe travel3m 23s
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Election polling methodologies2m 57s
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Is your data any good?3m 44s
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Storytelling with data2m 41s
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The middle of my data3m 21s
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The ubiquitous bell curve3m 23s
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Decoding polling results3m 27s
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What is an outlier?3m 15s
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Statistical bias3m 45s
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The importance of regression analysis3m 38s
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Understanding probabilities3m 23s
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Statistics tools3m 19s
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Modern organizations use statistics3m 16s
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Combinations3m 46s
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Measuring variation3m 55s
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Sample space3m 51s
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Election win probabilities3m 41s
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Cognitive bias4m
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Forecasting3m 13s
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Toilet paper4m 5s
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Winning streaks3m 6s
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Coffee3m 6s
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Vaccines2m 44s
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Superfans2m 50s
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US Presidents2m 51s
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The cost of owning a pet2m 45s
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Funny movies3m 23s
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Success in the music industry2m 23s
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Home Improvement3m 4s
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Youth sports3m 30s
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Mental health2m 43s
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Chocolate2m
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Baby statistics2m 30s
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Books2m 48s
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Commutes2m 43s
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Farms2m 15s
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Hip hop2m 50s
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Mass transit2m 43s
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Major league baseball3m 19s
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Placebos2m 54s
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Podcasts2m 13s
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Social media2m 44s
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Supply chains2m 51s
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