Question

imagine you toss a fair coin 300 times and note how many times heads turns up. simulating this experiment should work just as well if your experiment is correctly set up. one way of doing this is by using your spreadsheet program commands. open this spreadsheet and go to the fair coin tab. this will give you one sample set, which we will call sample 1. thus, you now have one sample proportion of heads. answer these questions based on this information. part a what is the value of the sample proportion of heads? what can you infer from this value?

Answer 1

The sample proportion of heads from a coin-toss experiment can be calculated by dividing the number of heads by the total tosses, and it reflects the experimental probability. The Law of Large Numbers indicates that this proportion should approach 50% for a fair coin as the number of trials increases.

Step-by-step explanation:

The sample proportion of heads in your experiment using a spreadsheet can be found by dividing the number of times heads appeared by the total number of tosses. For instance, if you obtained heads 150 out of 300 times, your sample proportion would be 150/300 = 0.5. This means that in your simulation, the observed frequency of heads is exactly the theoretical probability of a fair coin, which is 50%.

The Law of Large Numbers tells us that as the number of trials increases, the experimental probability (relative frequency) approaches the theoretical probability. Even though individual experiments may deviate from the expected 50% ratio of heads to tails, over many trials, the relative frequency should get closer to the theoretical value of 0.5 for a fair coin. Therefore, a sample proportion close to 0.5 in a large experiment like 300 tosses would support the fairness of the coin, while a substantial deviation might suggest a biased coin.

Answer 2

The sample proportion of heads from a fair coin tossed 300 times is an estimate of the true probability and should be close to 0.5, demonstrating the law of large numbers.

Step-by-step explanation:

The value of the sample proportion of heads in a simulation of tossing a fair coin 300 times represents the relative frequency with which heads come up in that particular sample. If the coin is truly fair, we would expect this proportion to be close to 0.5 (or 50%), given that the theoretical probability of flipping heads with a fair coin is 0.5. However, this proportion might not be exactly 0.5 in any single experiment due to random fluctuations, but by the law of large numbers, it would typically approach 0.5 as the number of trials increases.

Karl Pearson's experiment, where he tossed a coin 24,000 times, confirmed that the relative frequency of heads obtained was very close to the theoretical probability, highlighting the law of large numbers. Therefore, we can infer that the sample proportion of heads obtained from the 300 coin tosses provides an estimate of the true probability of getting heads from this coin and should be close to the long-term expected value of 0.5 if the coin is indeed fair.