Let’s Summarize

• The following concepts apply to all of the chi-square hypothesis tests in this module:
  • The chi-square distribution is a distribution that is skewed to the right.
  • The variability (or spread) of the chi-square distribution depends on the degrees of freedom of the distribution.
  • The test statistic for a chi-square distribution is always greater than or equal to zero.

• The following concepts apply for a chi-square goodness-of-fit test:
  • The null hypothesis is that the distribution fits the hypothesized proportions. The alternative hypothesis is that the distribution does not fit the hypothesized proportions.
  • Expected counts are found by taking the total count and multiplying by each of the hypothesized proportions.
  • The expected counts need to be 5 or more to conduct a chi-square test and are NOT rounded to the nearest whole number.
  • The degrees of freedom is [latex]k – 1[/latex], where [latex]k[/latex] is the number of categories.
  • The chi-square test statistic is the sum of [latex]\frac{(\mathrm{Observed} \ – \ \mathrm{Expected})^2}{\mathrm{Expected}}[/latex] for each category.

• The following concepts apply for a chi-square test of independence:
  • The null hypothesis is that there is no association between the two categorical variables. The alternative hypothesis is that there is an association between the two categorical variables.
  • The degrees of freedom is [latex](r – 1)(c – 1)[/latex], where r is the number of rows in the contingency table and [latex]c[/latex] is the number of columns in the contingency table.
  • The expected count for each cell is found by taking the row total times the column total and dividing it by the grand total.
  • The expected counts need to be 5 or more to conduct a chi-square test and are NOT rounded to the nearest whole number.
  • The chi-square test statistic is the sum of [latex]\frac{(\mathrm{Observed} \ – \ \mathrm{Expected})^2}{\mathrm{Expected}}[/latex] for each cell in the contingency table.

• The following concepts apply for a chi-square test of homogeneity:
  • The null hypothesis is that the distribution of the two populations is the same. The alternative hypothesis is that the distribution of the two populations is not the same.
  • The expected count for each cell is found by taking the row total times the column total and dividing it by the grand total.
  • The expected counts need to be 5 or more to conduct a chi-square test and are NOT rounded to the nearest whole number.
  • The degrees of freedom for a chi-square test of homogeneity for two populations is [latex]k – 1[/latex], where [latex]k[/latex] is the number of response values.
  • The chi-square test statistic is the sum of [latex]\frac{(\mathrm{Observed} \ – \ \mathrm{Expected})^2}{\mathrm{Expected}}[/latex] for each category.

To determine which type of chi-square test is being done, consider the number of samples and the general research question that is being answered.