Coin 1 is fair and coin 2 has probability of heads 3 / 4. A...

Coin 1 is fair and coin 2 has probability of heads $3 / 4$. A test involves flipping a coin repeatedly until the first occurrence of heads. The number of tosses is observed. (a) Can you design a test to determine whether the fair coin is in use? Assume $\alpha=5 \%$. What is the probability of detecting the biased coin? (b) Repeat part a if the biased coin has probability $1 / 4$.

Coin 1 is fair and coin 2 has probability of heads $3 / 4$. A test involves flipping a coin repeatedly until the first occurrence of heads. The number of tosses is observed.
(a) Can you design a test to determine whether the fair coin is in use? Assume $\alpha=5 \%$. What is the probability of detecting the biased coin?
(b) Repeat part a if the biased coin has probability $1 / 4$.

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Key Concepts

Hypothesis Testing

A framework for making decisions using data, where one sets up a null hypothesis representing a default position and an alternative hypothesis representing a claim one wishes to test, then uses statistical evidence to decide whether to reject the null hypothesis.

Significance Level (Alpha Level)

The pre-determined threshold for deciding whether an observed effect is statistically significant. It represents the probability of incorrectly rejecting the null hypothesis when it is actually true, often set at 5% in many tests.

Power of a Test

The probability that the test will correctly reject a false null hypothesis. It is a measure of the ability of a test to detect an effect when there is one, and depends on factors such as the true parameter, sample size, and chosen significance level.

Geometric Distribution

A discrete probability distribution that models the number of independent trials needed to achieve the first success in a sequence of Bernoulli (yes/no) trials. Its key property is defined by the constant probability of success on each trial, which makes it a useful model for sequential events like repeated coin flips until a head appears.

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