You are shown a coin that its owner says is fair in the sense that it will produce the same number of heads and tails when flipped a very large number of times.
Suppose you decide to flip a coin 100 times.
a. What conclusion would you be likely to draw if you observed 95 heads?
b. What conclusion would you be likely to draw if you observed 55 heads?
You are shown a coin that its owner says is fair in the sense that it will produce the same number of heads and tails when flipped a very large number of times.
a. Describe an experiment to test this claim.
b. What is the population in your experiment?
c. What is the parameter?
e. What is the statistic?
f. Describe briefly how statistical inference can be used to test the claim.
a - either you've just seen something that's phenomenally unlikely, or the coin's owner isn't being truthful about the fairness of the coin.
b - you've just seen something that's consistent with the coin's owner's pronouncement that the coin is fair. (Not that that means it definitely IS fair: it's just that you don't have enough evidence to conclude that it isn't.)
I suspect you're about to be introduced to the concept of the so-called "null hypothesis", which in this instance is that the coin is fair, i.e. that heads and tails are equally likely to occur. When you do an experiment, either you get a result which is consistent with your null hypothesis, or you get a result which is so unlikely if the nullhypothesis is true that you're forced to throw the null hypothesis away. Your two questions are examples of that process.
Thanks so much for the reply!
To answer these questions, we need to understand the concept of hypothesis testing and the role of p-values.
a. If you observe 95 heads out of 100 coin flips, it would be quite unusual under the assumption that the coin is fair. To draw a conclusion, you can perform a hypothesis test with the null hypothesis (H0) stating that the coin is fair and the alternative hypothesis (Ha) stating that the coin is biased.
In this case, you can use a binomial test or a chi-square test to calculate the probability of obtaining 95 or more heads if the coin is actually fair. This probability is known as the p-value. If the p-value is very small (below a predetermined significance level), such as 0.05, you would reject the null hypothesis and conclude that the coin is likely biased towards heads.
b. If you observe 55 heads out of 100 coin flips, this result is more consistent with a fair coin. Again, you can perform a hypothesis test using the same null and alternative hypotheses. By calculating the p-value, you can see if obtaining 55 or more heads is likely to occur by chance alone.
If the p-value is greater than the significance level (e.g., 0.05), you would fail to reject the null hypothesis and conclude that there is not enough evidence to suggest the coin is biased. In this case, the result is within the range of randomness expected from a fair coin.
In summary, the conclusion drawn from these observations is based on the calculated p-value. If the p-value is below the significance level, you reject the null hypothesis in favor of the alternative hypothesis, suggesting the coin is biased. If the p-value is above the significance level, you fail to reject the null hypothesis and conclude there is not enough evidence to support the alternative hypothesis.