Question

Design a random experiment and a hypothesis testing scheme to assess whether a given coin is fair or not. Explain all components of your solution in detail

Answer 1

A random experiment and a hypothesis testing scheme to assess whether a given coin is fair or not ;

the idea of hypothesis testing. This is a formal framework that we can use to pose questions about a variety of topics in a consistent form that lets us apply statistical techniques to make statements about how results that we've gathered relate to questions that we're interested in. If we carefully follow the rules of hypothesis testing, then we can confidently talk about the probability of events that we've observed under a variety of hypotheses, and put our faith in those hypotheses that seem the most reasonable. Central to the idea of hypothesis testing is the notion of null and alternative hypotheses. We want to present our question in the form of a statement that assumes that nothing has happened (the null hypothesis) versus a statement that describes a specific relation or condition that might better explain our data.For example, suppose we've got a coin, and we want to find out if it's true, that is, if, when we flip the coin, are we as likely to see heads as we are to see tails.

A null hypothesis for this situation could be

H0: We're just as likely to get heads as tails when we flip the coin.

A suitable alternative hypothesis might be:

Ha: We're more likely to see either heads or tails when we flip the coin.

An alternative hypothesis such as this is called two-sided, since we'll reject the null hypothesis if heads are more likely or if tails are more likely. We could use a one-sided alternative hypothesis like "We're more likely to see tails than heads," but unless you've got a good reason to believe that it's absolutely impossible to see more heads than tails, we usually stick to two-sided alternative hypotheses.Now we need to perform an experiment in order to help test our hypothesis. Obviously, tossing the coin once or twice and counting up the results won't help very much. Suppose I say that in order to test the null hypothesis that heads are just as likely as tails, I'm going to toss the coin 100 times and record the results. Carrying out this experiment, I find that I saw 55 heads and 45 tails. Can I safely say that the coin is true? Without some further guidelines, it would be very hard to say if this deviation from 50/50 really provides much evidence one way or the other. To proceed any further, we have to have some notion of what we'd expect to see in the long run if the null hypothesis was fair . Then, we can come up with some rule to use in deciding if the coin is fair or not, depending on how willing we are to make a mistake. Sadly, we can't make statements with complete certainty, because there's always a chance that, even if the coin was fair, we'd happen to see more heads than tails, or, conversely, if the coin was weighted, we might just happen to see nearly equal numbers of heads and tails when we tossed the coin many times. The way we come up with our rule is by stating some assumptions about what we'd expect to see if the null hypothesis was true, and then making a decision rule based on those assumptions. For tossing a fair coin (which is what the null hypothesis states), most statisticians agree that the number of heads (or tails) that we would expect follows what is called a binomial distribution. This distribution takes two parameters: the theoretical probability of the event in question (let's say the event of getting a head when we toss the coin), and the number of times we toss the coin. Under the null hypothesis, the probability is .5. In R, we can see the expected probability of getting any particular number of heads if we tossed a coin 100 times by plotting the density function for the binomial distribution with parameters 100 and .5 .

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