Glossary
Alternative Hypothesis (Hₐ)
A statement that contradicts the null hypothesis, suggesting that the observed and expected distributions are significantly different. It's what the researcher is trying to find evidence for.
Example:
If testing a new fertilizer, the Alternative Hypothesis (Hₐ) might state that plants treated with the fertilizer will grow significantly taller than those without.
Chi-Square Goodness of Fit Test
A statistical test used to determine if an observed frequency distribution matches a theoretical expected distribution. It assesses whether differences between observed and expected data are statistically significant.
Example:
A candy company claims its bags have a specific color distribution; a Chi-Square Goodness of Fit Test could check if a sampled bag's colors match that claim.
Chi-Square Statistic (χ²)
A calculated value that quantifies the discrepancy between observed and expected frequencies in a chi-square test. A larger value indicates a greater difference between the observed and expected data.
Example:
After surveying student preferences for school lunch, a high Chi-Square Statistic (χ²) value would suggest that the observed preferences are very different from what was expected.
Context (in conclusion)
The practice of relating the statistical findings of a hypothesis test back to the specific real-world scenario or problem being investigated. It ensures the conclusion is meaningful and understandable.
Example:
Instead of just saying 'Reject H₀,' a good conclusion includes context by stating, 'We reject the null hypothesis, providing convincing evidence that the distribution of car colors in the city is different from the national distribution.'
Critical Value
A threshold value from a statistical distribution (like the chi-square distribution) that is compared to the test statistic to make a decision about the null hypothesis. If the test statistic exceeds this value, the null hypothesis is rejected.
Example:
If your calculated chi-square statistic is 12 and the Critical Value for your test is 9.488, you would reject the null hypothesis.
Degrees of Freedom (df)
A value that indicates the number of independent pieces of information used to calculate a statistic. For a chi-square goodness of fit test, it's the number of categories minus one.
Example:
If you're analyzing the distribution of 5 different car colors, the Degrees of Freedom (df) would be 5 - 1 = 4.
Expected distribution
The theoretical counts or frequencies for each category that would be anticipated if the null hypothesis were true. These are the 'Expected' values in the chi-square formula, calculated based on a hypothesized proportion or uniform distribution.
Example:
If a fair six-sided die is rolled 60 times, the expected distribution for each number (1-6) would be 10 rolls.
Fail to reject H₀
The decision made when the p-value is greater than or equal to the significance level (α), or the test statistic is less than or equal to the critical value. This indicates there is not convincing evidence to support the alternative hypothesis.
Example:
If a study on a new teaching method shows no significant improvement (p-value > 0.05), you would Fail to reject H₀, meaning there's no strong evidence the new method is better.
Null Hypothesis (H₀)
A statement that there is no significant difference between the observed and expected distributions, or that the observed distribution is the same as the expected distribution. It's the statement assumed to be true until evidence suggests otherwise.
Example:
For a coin flip, the Null Hypothesis (H₀) would state that the coin is fair, meaning the proportion of heads is 0.5.
Observed frequency distribution
The actual counts or frequencies of outcomes recorded from a sample or experiment. These are the 'Observed' values in the chi-square formula.
Example:
In a survey of 100 students, finding that 30 prefer apples, 45 prefer bananas, and 25 prefer oranges represents the observed frequency distribution of fruit preferences.
P-value
The probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true. A small p-value suggests strong evidence against the null hypothesis.
Example:
A P-value of 0.01 means there's only a 1% chance of seeing your observed data (or more extreme) if the null hypothesis were true, leading to rejection.
Reject H₀
The decision made when the p-value is less than the significance level (α), or the test statistic exceeds the critical value. This indicates there is convincing evidence to support the alternative hypothesis.
Example:
If a new drug significantly outperforms a placebo with a p-value < 0.05, you would Reject H₀, concluding the drug is effective.
Significance Level (α)
The predetermined threshold probability used to decide whether to reject the null hypothesis. It represents the maximum probability of making a Type I error (rejecting a true null hypothesis).
Example:
Setting a Significance Level (α) of 0.05 means there's a 5% chance of incorrectly rejecting the null hypothesis if it's actually true.