8.1: Introducing Statistics: Are My Results Unexpected?
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Random Chance or Incorrect Claim?
Just as with any outcome, there is always a chance that something abnormal happens. The tricky part (AKA statistics) comes into play when we determine if our outcome was just due to a random chance or something incorrect in the original claim.
For instance, if we flip a fair coin 10 times, we would expect to get 5 heads and 5 tails. Would this absolutely be our results? Probably not. Flipping a coin is a random process that could result in a variety of outcomes. The most likely outcome would be 5 heads and 5 tails, but there are other outcomes that are almost just as likely.
If we flipped a coin 10 times and got 4 heads and 6 tails, would we doubt that the coin was a fair coin? Probably not. That is a normal outcome and it is pretty close to our expected counts of 5 and 5. If we were to get 10 heads and 0 tails, this is a much larger discrepancy so this might cause us to doubt that the coin is really a fair coin.
Just like we had with our other inference procedures, our sample size plays a huge part in our outcome. When we flip a coin 10 times and receive 4 heads and 6 tails, no big deal. If we flipped a coin 1000 times and received 400 heads and 600 tails, that seems a lot more unlikely.
The main reason why the sample size affects our expected outcome is due to the standard deviation decreasing🠋 as the sample size increases🠉. This is a very important relationship among all statistics that we have discussed this year and is almost sure to show up on the AP exam several times.
Law of Large Numbers
The law of large numbers states that as we perform a simulation more and more times, our outcome gets closer and closer to the true, expected numbers. In this case, as we flip a coin more and more times, we would expect the number of heads and tails to get closer to a 50/50 split (as it would).
Therefore, if we flipped a coin 1000 times, while it may not be 500 heads and 500 tails, we would expect the percentages to be very close to 50/50. Therefore, if we still had a 40/60 count after 1000 flips, we might start to doubt the fairness of the coin. As previously stated, the law of large numbers applies because we decrease our standard deviation by increasing sample size, which in turn decreases our margin of error and hones in on the true counts more accurately.