Unit Eight: Inference for Categorical Data (Chi-Square)- essential knowledge

What types of variation can occur between what we observe and what we expect?

What types of variation can occur between what we observe and what we expect?

Variation can be either random or non-random, which can indicate whether the observed results deviate due to chance or a meaningful effect.

What are expected counts in the context of categorical data?

Expected counts are the counts consistent with the null hypothesis and are generally calculated as the sample size multiplied by the probability for each category.

What does the chi-square statistic measure?

The chi-square statistic measures the distance between observed and expected counts relative to the expected counts, helping to determine if there is a significant difference.

What are the key characteristics of chi-square distributions?

Chi-square distributions have only positive values and are right-skewed, with the skew decreasing as the degrees of freedom increase.

What are the hypotheses for a chi-square goodness-of-fit test?

The null hypothesis specifies the expected proportions for each category, while the alternative hypothesis states that at least one proportion is different from what is specified in the null hypothesis.

When is a chi-square test for goodness of fit used?

It is used when testing the distribution of proportions for one categorical variable against expected proportions.

How do you calculate expected counts for a chi-square goodness-of-fit test?

Expected counts are calculated as (sample size) × (null proportion) for each category.

What is the null distribution in a chi-square test?

The null distribution is either a randomization distribution or, when assuming a probability model is true, a theoretical chi-square distribution.

How do you find the p-value for a chi-square goodness-of-fit test?

The p-value can be found using a chi-square table or computer-generated output based on the test statistic and degrees of freedom.

How should the p-value for a chi-square goodness-of-fit test be interpreted?

The p-value represents the probability, assuming the null hypothesis is true, of obtaining a test statistic as extreme or more extreme than the observed value.

What is the decision rule for a chi-square goodness-of-fit test?

Compare the p-value to the significance level (α). If p ≤ α, reject the null hypothesis; if p > α, fail to reject the null hypothesis.

How can the results of a chi-square goodness-of-fit test be used in research?

The results provide statistical reasoning to support or refute an answer to a research question regarding the sampled population.

What are the hypotheses for a chi-square test for homogeneity?

H0: There is no difference in the distributions of a categorical variable across populations or treatments. Ha: There is a difference in these distributions.

What are the hypotheses for a chi-square test for independence?

H0: There is no association between two categorical variables in the population (they are independent). Ha: The variables are associated (dependent).

When is a chi-square test for homogeneity appropriate?

It is used when comparing distributions to see if proportions across different populations are the same.

When is a chi-square test for independence appropriate?

It is used to determine if there is an association between two categorical variables in a population, based on a two-way table of categorical data.

How do you find the p-value for a chi-square test for independence or homogeneity?

The p-value can be found using a chi-square table or technology, considering the test statistic and degrees of freedom.

How is the p-value for a chi-square test for independence or homogeneity determined?

It is the proportion of values in the chi-square distribution with the same or more extreme test statistics than the one calculated.

How should the p-value for a chi-square test for homogeneity or independence be interpreted?

The p-value represents the probability, assuming the null hypothesis is true, of obtaining a test statistic as extreme or more extreme than the observed value.

What is the decision rule for a chi-square test for homogeneity or independence?

Compare the p-value to the significance level (α). If p ≤ α, reject the null hypothesis; if p > α, fail to reject the null hypothesis.

How can the results of a chi-square test for homogeneity or independence be used in research?

The results provide statistical reasoning to answer research questions about either the sampled population (independence) or the populations being compared (homogeneity).