If a coin is tossed 100 times, we would expect approximately...

If a coin is tossed 100 times, we would expect approximately 50 of the outcomes to be heads. It can be demonstrated that a coin is unfair if $h,$ the number of outcomes that result in heads, satisfies $\left|\frac{h-50}{5}\right| \geq 1.645$ . Describe the number of outcomes that determine an unfair coin that is tossed 100 times.

Key Concepts

Hypothesis Testing

Hypothesis testing is a formal method for deciding whether observed data significantly deviates from what is expected under a specific assumption, such as the fairness of a coin. By comparing the calculated statistic (like a z-score) against a predetermined threshold, one can decide whether to reject the null hypothesis of fairness.

Z-Score

A z-score quantifies how many standard deviations an observed value is away from the mean. In the context of evaluating coin toss results, the z-score is used to determine the distance of the observed number of heads from the expected mean, helping to decide whether the deviation is statistically significant.

Critical Values

Critical values define the boundaries of the acceptance region in a hypothesis test. In the coin tossing scenario, if the standardized deviation of the number of heads falls beyond the critical z-value, it indicates that the coin's behavior differs significantly from what is expected under fairness, suggesting that the coin may be biased.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent experiments, each with two possible outcomes and a constant probability of success. In the context of coin tosses, each toss is a Bernoulli trial where getting heads is considered a success, and the total number of heads out of 100 tosses follows a binomial distribution.

Normal Approximation

For a large number of trials, the binomial distribution can be approximated by a normal distribution, thanks to the Central Limit Theorem. This approximation allows the use of continuous probability tools, such as z-scores, to evaluate the probabilities of observing extreme deviations from the mean in coin tossing experiments.

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