Question

Students at the Akademia Podlaka conducted an experiment to determine whether the Belgium-minted Euro coin was equally likely to land heads up or tails up. Coins were spun on a smooth surface, and in 250 spins, 140 landed with the heads side up (New Scientist, January 4 , 2002). Should the students interpret this result as convincing evidence that the proportion of the time the coin would land heads up is not .5? Test the relevant hypotheses using \(\alpha=.01\). Would your conclusion be different if a significance level of \(.05\) had been used? Explain.

Short answer

The short answer will depend on the calculated P-value and its comparison with the provided significance levels. It will be of the form: 'For \(\alpha = 0.01\), the evidence is insufficient (or sufficient) to say the coin is unfair. However, for \(\alpha = 0.05\), we reject (or do not reject) the null hypothesis indicating the coin is (or isn't) unfair.'

Step-by-step solution

State the null and alternate hypotheses

The Null Hypothesis Ho : p = 0.5 (The coin is fair - probability of landing heads is 0.5). The Alternate Hypotheses Ha : p ≠ 0.5 (The coin is not fair - probability of landing heads is not 0.5)

Determine the Test Statistics

The test statistic for a hypothesis test for a proportion is a z-score (z). The formula for the z- score is \[ z = (x - np_0) / \sqrt{ np_0(1 - p_0)} \] where n is the number of trials, \(p_0\) is the expected probability under the null hypothesis and x is the number of successes. Here, n = 250 (number of spins), \(p_0 = 0.5\) and x = 140 (coins landing heads). Plugging these in the equation will give the z score.

Calculate the P-value

The P-value is the probability that a z-score is as extreme as, or more extreme than, the observed z-score, assuming the null hypothesis is true. It can be found using the standard normal distribution (z-distribution) and statistical tables or software. It is calculated for both sides of the distribution (because the alternate hypothesis is p ≠ 0.5), hence it is twice the probability of the tail beyond the test statistic.

Compare the P-value to the predefined significance level

Compare the P-value with the provided significance levels (\(\alpha = 0.01\) and \( \alpha = 0.05\)). If the P-value is less than or equal to \(\alpha\), reject the null hypothesis. In this case, conclude the coin is not fair.

Interpret the result

There are two conclusions to be reached - one for each significance level. If for \(\alpha = 0.01\), the null hypothesis is not rejected, there is insufficient evidence to conclude the coin is unfair based on this level. If for \(\alpha = 0.05\), the null hypothesis is rejected, conclude the coin is unfair based on this level. The conclusion may change based on the chosen significance level.