2.3 Statistics for Two Categorical Variables

• Categorical variables are variables whose values can be grouped into distinct categories or classes
• What data is needed to create a contingency table?

  Two categorical variables
• Conditional probability is the probability of an event occurring given that another event has already occurred
• What is a contingency table used for?

  Organize data on categorical variables
• A contingency table can only have two categories for each variable

  False
• What is the first step in testing for statistical independence using a contingency table?

  Calculate conditional probabilities
• Steps to calculate expected frequencies

  1️⃣ Identify the row and column totals

  2️⃣ Multiply the row and column totals

  3️⃣ Divide the result by the grand total
• Steps to calculate the expected frequency for males who like math in the given example:

  1️⃣ Identify the row total for males

  2️⃣ Identify the column total for likes math

  3️⃣ Identify the grand total

  4️⃣ Apply the formula: (Row Total × Column Total) / Grand Total
• What is the purpose of a contingency table?

  Organize categorical data
• The probability that a student is male given they prefer math is calculated as P(Male|Math) = 45 / 95
• What is the probability that a male likes math, given the data in the contingency table?

  0.45
• What do expected frequencies represent in a contingency table?

  Frequencies under independence
• Expected frequencies are calculated under the assumption that categorical variables have no relationship.


  True
• The Chi-Square test for independence assesses whether two categorical variables are statistically independent.

  True
• How are degrees of freedom calculated in the Chi-Square test?

  (Rows - 1) × (Columns - 1)
• With \( df = 2 \) and a p-value < 0.05, we conclude that gender and preferred subject are dependent.


  True
• Match the type of categorical variable with its characteristic:

  Nominal ↔️ No inherent order

  Ordinal ↔️ Categories have a specific order
• What is the formula for conditional probability denoted as?

  P(A|B)
• What is the purpose of a contingency table?

  Organize categorical data
• Steps to create a contingency table

  1️⃣ Identify the two categorical variables

  2️⃣ List the categories for rows and columns

  3️⃣ Record the count in each cell

  4️⃣ Include row and column totals
• Two categorical variables are statistically independent if P(A|B) = P(A)

  True
• The formula to calculate expected frequency for a cell is (Row Total × Column Total) / Grand Total
• Expected frequencies represent the hypothetical number of observations in a contingency table if the variables are statistically independent.


  True
• Match the type of categorical variable with its characteristic:

  Nominal ↔️ No inherent order

  Ordinal ↔️ Specific order or ranking
• What does the notation P(A|B) represent in conditional probability?

  Probability of A given B
• In the example provided, gender and math preference are statistically dependent
• Gender and math preference are statistically independent based on the given probabilities.

  False
• The denominator in the formula for calculating expected frequencies is the Grand Total.
• The expected frequency for females who like English is 35.
• Observed frequencies in the Chi-Square test are denoted as \( O_{ij} \).

  True
• What is the approximate value of the Chi-Square test statistic in the example provided?

  10.54
• Nominal variables have categories that are distinct but have no inherent order
• A contingency table shows how many male and female students prefer subjects
• Match the categorical variable type with its characteristic:

  Nominal ↔️ No inherent order

  Ordinal ↔️ Specific order or ranking
• What is the first step in creating a contingency table?

  Identify the two categorical variables
• To compute conditional probabilities from a contingency table, you divide the cell count by the total count of the condition
• What do expected frequencies represent in a contingency table?

  Frequencies under independence
• The formula to calculate the expected frequency for a cell is (Row Total × Column Total) / Grand Total
• Nominal variables have categories with no inherent order
• Steps to create a contingency table:

  1️⃣ Identify the two categorical variables

  2️⃣ List the categories for each variable as rows and columns

  3️⃣ Record the count for each cell

  4️⃣ Include row and column totals