91Ó°ÊÓ

In probability and statistics, the binomial distribution is a common way to model scenarios where there are two possible outcomes, often termed as "success" or "failure." It is used to calculate the probabilities of obtaining a fixed number of "successes" in a specific number of trials. In the context of our euro coin problem:

• The trial is each spin of the coin.
• The outcome "success" can be defined as the coin landing on heads.

Here, we're interested in the probability of achieving 140 heads in 250 spins. The variables for this binomial distribution include:

• Number of trials, denoted as \(n\), which equals 250.
• Probability of success on each trial, denoted as \(p\), which should be 0.5 if the coin is fair.

With these, the expected number of successes or heads is calculated by multiplying the number of trials by the probability of success: \(n \times p = 250 \times 0.5 = 125\). Truly understanding the mechanics of a binomial distribution helps in assessing how likely or unlikely it is for an event, like 140 heads, to occur.

The Z-score is a statistical measurement that tells you how many standard deviations an element is away from the mean. In our case, it helps determine how far the observed number of heads, 140, deviates from what we'd expect if the coin were fair, which is 125.

• First, calculate the standard deviation for our binomial distribution. The formula is: \(\sigma = \sqrt{n \times p \times (1-p)}\).
• Here, \(n = 250\) and \(p = 0.5\), leading to \(\sigma \approx 7.91\).

Next, use the Z-score formula: \(Z = \frac{\text{Observed} - \text{Expected}}{\sigma}\). Plugging in our values:

• Observed = 140
• Expected = 125

We find \(Z = \frac{140 - 125}{7.91} \approx 1.90\). This Z-score tells us the degree of deviation from the mean in terms of standard deviations, helping us decide the fairness of the euro based on normed expectations from the binomial distribution.

Hypothesis testing begins by formulating two contrasting hypotheses. The null hypothesis (\(H_0\)) is a statement of no effect or no difference. In this circumstance, it assumes that the euro coin is fair, meaning half the spins should result in heads and half in tails. Mathematically, this is represented as:

• \(H_0: p = 0.5\)

The alternative hypothesis (\(H_1\)), on the other hand, proposes that there is a deviation from what the null hypothesis states. Here, it suggests that the euro coin's probability of showing heads isn't 0.5:

• \(H_1: p eq 0.5\)

Typically, a statistical test (in this case, a Z-test) is used to gather evidence against the null hypothesis. If the evidence significantly contradicts \(H_0\) (based on a pre-determined significance level, often 0.05), \(H_0\) is rejected in favor of \(H_1\). In the case of our observation – the p-value being 0.057 – it is decided not to reject \(H_0\), as it’s slightly above the typical 0.05 threshold, meaning there's not enough evidence to consider the coin biased.