Carlos and Michael play a game where each flips a coin once:...

Carlos and Michael play a game where each flips a coin once: If the outcomes of the tosses are the same, then no one wins; but if the outcome is different the player with "heads" wiss Michael uses a fair coin but he suspects that Carlos is using a biased coin. (a) Find a $10 \%$ significance level test for an experiment that counts how many times los wins in 6 games to test whether Carlos is cheating. Repeat for $n=12$ games. (b) Now design a $10 \%$ significance level test based on the number of times Carlof tosses come up heads. Which test is more effective? (c) Find the probability of detection if Carlos uses a coin with $p=0.75 ; p=0.55$.

Carlos and Michael play a game where each flips a coin once: If the outcomes of the tosses are the same, then no one wins; but if the outcome is different the player with "heads" wiss Michael uses a fair coin but he suspects that Carlos is using a biased coin.
(a) Find a $10 \%$ significance level test for an experiment that counts how many times los wins in 6 games to test whether Carlos is cheating. Repeat for $n=12$ games.
(b) Now design a $10 \%$ significance level test based on the number of times Carlof tosses come up heads. Which test is more effective?
(c) Find the probability of detection if Carlos uses a coin with $p=0.75 ; p=0.55$.

Key Concepts

Critical Region

The critical region is the set of outcomes for a test statistic that leads to rejection of the null hypothesis. It is determined by the significance level and the probability distribution appropriate to the test (such as the binomial distribution in coin flip experiments), and it defines the cutoff point beyond which the observed results are deemed statistically unlikely under the null hypothesis.

Power of a Test

The power of a test is the probability that the test correctly rejects the null hypothesis when a specific alternative hypothesis is true. In other words, it measures the test’s effectiveness at detecting an effect (like bias in a coin) when it actually exists. A more effective test has a higher power, meaning it is more likely to flag true biased behavior.

Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It plays a key role in coin toss experiments, as each coin flip is an independent trial with two outcomes, and it is used to calculate probabilities and critical values in testing.

Significance Level

The significance level is the threshold for rejecting the null hypothesis; it quantifies the probability of making a Type I error, which is the error of rejecting the null hypothesis when it is actually true. In this context, a 10% significance level means that the test is designed to allow a 10% chance of incorrectly concluding that a fair coin is biased.

Null and Alternative Hypotheses

These are the two competing statements in a statistical test. The null hypothesis usually represents a standard or default claim (for example, that a coin is fair), while the alternative hypothesis represents a claim of deviation or effect (such as the coin being biased). The goal is to determine whether there is enough evidence in the sample data to favor the alternative hypothesis over the null hypothesis.

Hypothesis Testing

Hypothesis testing is the framework used to decide whether observed data provide sufficient evidence to reject a default position (the null hypothesis). It involves comparing the observed outcomes with what would be expected under the assumption that the null hypothesis is true, and then making a decision based on a predetermined significance level.

Please give Ace some feedback