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Explain the concept of 'variation' in chi-square tests.
Variation refers to the difference between observed and expected counts. Chi-square tests assess whether this variation is due to random chance or indicates a significant relationship.
Explain how sample size affects the standard deviation.
As sample size increases, standard deviation decreases. This inverse relationship is crucial for statistical inference.
Explain the Law of Large Numbers.
As sample size increases, the sample mean gets closer to the true population mean.
Explain the impact of a low p-value on the null hypothesis.
A low p-value (e.g., < 0.05) suggests that the observed difference is unlikely due to chance, leading to the rejection of the null hypothesis.
Explain the impact of a high p-value on the null hypothesis.
A high p-value (e.g., > 0.05) suggests that the observed difference could easily be due to chance, leading to a failure to reject the null hypothesis.
Explain how increasing the sample size affects the power of a statistical test.
Larger sample sizes increase the power of a test because they lead to smaller standard deviations, making it easier to detect a true effect if it exists.
What are the differences between statistical significance and practical significance?
Statistical Significance: A result is statistically significant if the p-value is below the significance level. | Practical Significance: Considers whether the magnitude of the effect is meaningful in the real world, regardless of statistical significance.
What are the differences between a Chi-Square Goodness-of-Fit test and a Chi-Square Test for Independence?
Goodness-of-Fit: Tests if the distribution of a categorical variable matches a claimed distribution. | Test for Independence: Tests if there is an association between two categorical variables.
What are the differences between the null and alternative hypotheses?
Null Hypothesis: A statement of no effect or no difference. | Alternative Hypothesis: A statement that contradicts the null hypothesis, suggesting there is a significant effect or difference.
What are the differences between Type I and Type II error?
Type I error: Rejecting the null hypothesis when it is true (false positive). | Type II error: Failing to reject the null hypothesis when it is false (false negative).
Define 'p-value'.
The probability of observing results as extreme as, or more extreme than, the observed data if the null hypothesis is true.
What is the null hypothesis?
A statement of no effect or no difference, which we aim to disprove with statistical evidence.
Define 'alternative hypothesis'.
A statement that contradicts the null hypothesis, suggesting there is a significant effect or difference.
What is 'statistical significance'?
A result is statistically significant if the p-value is less than the significance level (alpha), indicating the observed effect is unlikely due to chance.
Define 'degrees of freedom' in a Chi-Square test.
The number of independent pieces of information used to calculate the test statistic. For goodness-of-fit, df = (number of categories - 1).
What is 'power' in a statistical test?
The probability of correctly rejecting the null hypothesis when it is false.