Question

Alan, while snooping around his grandmother's basement stumbled upon a shiny object protruding from under a stack of boxes . When he reached for the object a genie miraculously materialized and stated: "You have found my magic coin. If you flip this coin an infinite number of times you will notice that heads will show \(60 \%\) of the time." Soon after the genie's declaration he vanished, never to be seen again. Alan, excited about his new magical discovery, approached his friend Ken and told him about what he had found. Ken was skeptical of his friend's story, however, he told Alan to flip the coin 100 times and to record how many flips resulted with heads. (a) What is the probability that Alan will be able convince Ken that his coin has special powers by finding a p value below 0.05 (one tailed). Use the Binomial Calculator (and some trial and error) (b) If Ken told Alan to flip the coin only 20 times, what is the probability that Alan will not be able to convince Ken (by failing to reject the null hypothesis at the 0.05 level \() ?\)

Short answer

(a) Alan has a high probability of convincing Ken if he gets 58+ heads in 100 flips. (b) Alan is unlikely to convince Ken with only 20 flips.

Step-by-step solution

Define the Problem Context

Alan has a magical coin that shows heads 60% of the time. His goal is to convince Ken that this is true by flipping the coin and getting a significant result that reje ts the null hypothesis (that the coin is fair with p=0.5) at a 0.05 significance level.

Identify Parameters for Part (a)

For part (a), Alan flips the coin 100 times. Here, we need to calculate the probability of getting a number of heads that gives a p-value less than 0.05 (one-tailed test) under the null hypothesis that the success probability per flip is \(p = 0.5\).

Calculate the Critical Value for Part (a)

Using the binomial distribution with \(n = 100\) and \(p = 0.5\), find the number of heads corresponding to a cumulative probability of 0.95. Here, the goal is to find the minimum number of heads, \(x\), for which \(P(X \geq x) < 0.05\).

Apply the Binomial Calculator for Part (a)

Trial and error with a binomial calculator indicates that if Alan gets 58 or more heads, the p-value will be below 0.05, thus convincing Ken. The probability of getting at least 58 heads by flipping a coin 100 times with the probability of heads being 60% is \(1 - P(X < 58)\) where \(X \sim B(100, 0.6)\).

Identify Parameters for Part (b)

For part (b), Alan flips the coin only 20 times. We must calculate the probability of getting fewer heads than necessary to reject the null hypothesis (i.e., \(P(X \leq x'\)) where \(x'\) results in failing to reject) where \(X \sim B(20, 0.6)\).

Calculate the Critical Value for Part (b)

Using the binomial distribution with \(n = 20\) and \(p = 0.5\), find the critical number of heads for rejecting the null hypothesis (\(P(X \geq x) < 0.05\)). This gives us \(x\) values where we wouldn't reject the null hypothesis.

Apply the Binomial Calculator for Part (b)

With the binomial calculator, one finds that Alan needs less than 14 heads to not reject the null hypothesis, under a fair coin (p-value > 0.05). Compute \(P(X \leq 13)\) where \(X \sim B(20, 0.6)\).

Conclusion for Part (a) and (b)

For part (a), \(P(X \geq 58) = 0.964\), confirming there is a high likelihood of convincing Ken. For part (b), \(P(X \leq 13) = \) approximately 0.952, indicating a high chance of failing to convincingly reject the null hypothesis.

Key concepts

Hypothesis Testing

Hypothesis testing is a fundamental aspect of statistical analysis. It allows us to make decisions or inferences about a population based on sample data. In Alan's coin scenario, hypothesis testing helps determine whether the coin is truly biased or just appears to be due to random chance.
The process involves two primary hypotheses: the null hypothesis and the alternative hypothesis. The null hypothesis (often denoted as \(H_0\)) is a statement of no effect or no difference. For Alan, \(H_0\) posits that the coin is fair, meaning it shows heads 50% of the time.
The next step is to test this hypothesis by collecting data, such as flipping the coin a number of times. After collecting the data, a statistical test, in this case, a binomial test, helps determine whether the results significantly deviate from what the null hypothesis predicts. This is where Alan is trying to convince Ken by showing significant results. A significant result indicates that the null hypothesis is unlikely given the observed data, thus providing support for the alternative hypothesis.

Significance Level

The significance level is a threshold set by the researcher, which indicates how extreme the results must be to reject the null hypothesis. The common significance level used in hypothesis testing is 0.05, though it can be set lower or higher depending on the context.
In Alan's experiment, a 0.05 significance level means there's a 5% risk of concluding that the coin is biased when it's actually not (a false positive). If the calculated p-value from the test is below this significance level, Alan can confidently reject the null hypothesis, suggesting his coin may indeed be special.

• A p-value less than 0.05 generally indicates strong evidence against the null hypothesis.
• If the p-value is greater than 0.05, the evidence is insufficient to reject the null hypothesis.

The significance level acts as a safeguard against accepting spurious results as genuine findings.

Null Hypothesis

The null hypothesis plays a crucial role in hypothesis testing. It acts as a default or baseline statement to compare against the researcher's actual claim. In many cases, the null hypothesis represents the skeptic's view, assuming no effect or no difference exists.
For Alan, the null hypothesis is that the coin is fair, meaning the probability of flipping heads is equal to that of tails (\(p = 0.5\)). This assumption sets the stage for testing whether there is sufficient evidence to claim that the coin behaves otherwise.
The goal is to collect data and examine whether the outcome provides a significant reason to reject this hypothesis. If the data aligns too closely with the null hypothesis, like in scenario (b) of the exercise where Alan is less likely to reject the null hypothesis with only 20 flips, then it suggests that any observed deviations could be due to chance rather than a special property of the coin.

Probability Distribution

Probability distribution is a statistical function that describes all the possible values and probabilities of a random variable. In Alan's coin flipping exercise, the binomial distribution is employed to model the randomness of the outcome.
A binomial distribution is appropriate here because it deals with situations where there are fixed numbers of independent trials, each with two possible outcomes (like flipping a coin). For Alan's magical coin, each flip, which results in a head or a tail, is an independent event. The probability distribution helps calculate the likelihood of observing a certain number of heads in a specified number of flips.

• In part (a), using 100 flips provides a wider range of possible outcomes where Alan might witness significant deviation from the expected fair coin result, allowing him to evaluate how likely achieving such a number is under the null assumption.
• In part (b), with only 20 flips, the distribution of results aligns more closely with the null hypothesis, providing fewer chances to observe significant deviations purely by chance.

Probability distributions such as these are essential for understanding and predicting random events' behavior.