91Ó°ÊÓ

Understanding the null hypothesis (\( H_0 \) and the alternative hypothesis (\( H_1 \) is fundamental to any hypothesis testing, including the chi-square test for goodness of fit. In simple terms, the null hypothesis is the default assumption that there is no significant effect or difference in the data. For our coin-tossing experiment, the null hypothesis would claim the coin is fair, meaning that the probability of landing heads or tails is equally likely, thus both having a probability of 0.5.

The alternative hypothesis contradicts the null hypothesis. It proposes that there is an effect or a difference. In the context of the coin, the alternative hypothesis suggests that the coin is biased - that the probabilities of heads and tails are not equal. When we perform a chi-square test, we essentially are checking if the data is significantly different from what we would expect under the null hypothesis to support the alternative hypothesis.

The expected counts are an integral part of the chi-square test. They represent the frequencies we would anticipate in each category if the null hypothesis were true. Calculating the expected counts requires knowing the total sample size and the hypothesized probability of outcomes. For the coin, given it is presumed fair under the null hypothesis, we expect a 50-50 split between heads and tails from the total spins.

To calculate the expected counts, multiply the total number of trials (spins) by the expected probability of each outcome. If we spun the coin 50 times, and the expected probability of both heads and tails is 0.5, we’d expect to see (\( 50 \times 0.5 = 25 \) heads and 25 tails. In our exercise, the actual counts were 15 heads and 35 tails, which diverges from the expected counts.

The chi-square test statistic is a measure of how much the observed counts deviate from the expected counts. It is calculated using the formula: \( \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \), where \( O_i \) is the observed count and \( E_i \) is the expected count for category \( i \).

In our exercise, the chi-square test statistic is determined by taking the square of the difference between the observed and expected counts for both heads and tails, and then dividing by the expected count for each. The two results are then summed to give the chi-square test statistic. A higher chi-square statistic indicates a greater divergence between observed and expected results and suggests the null hypothesis may not hold true.

The critical chi-square value is a benchmark against which the calculated chi-square statistic is compared to determine the outcome of the hypothesis test. This critical value is based on the level of significance (often denoted as \( \alpha \) and commonly set at 0.05 for a 5% significance level) and the degrees of freedom, which is dependent on the number of categories being analyzed minus one.

In our scenario, where there are two categories (heads and tails), the degrees of freedom is \( 1 \) (since \( 2 - 1 = 1 \) categories). Given a significance level of 0.05 and one degree of freedom, the critical value from chi-square distribution tables is approximately 3.841. If our computed chi-square statistic exceeds this critical value, we reject the null hypothesis, concluding that there’s a significant difference – in this case, supporting the claim that the coin is biased.