Solved Exercises on Two-Way Tables in Grade 8

Master two-way tables: frequency, relative frequency, and data interpretation through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Completing a Two-Way Table
Exercise 1
Complete the two-way table showing pet ownership by gender:
DogCatTotal
Male12820
Female15?25
Total27?45
Definition:

Two-Way Table: A table that organizes data by two categories to show the relationship between them

Table completion method:
  1. Identify what values are missing
  2. Use the fact that rows and columns must sum correctly
  3. Apply the principle that totals must match
Known Values
Male Cats = 8, Female Total = 25
Row Calculation
Female Cats = 25 - 15 = 10
Column Total
Total Cats = 8 + 10 = 18
Step 1: Find Female Cat owners

Female total = Female Dog + Female Cat owners

25 = 15 + Female Cat owners

Female Cat owners = 25 - 15 = 10

Step 2: Find Total Cat owners

Total Cat = Male Cat + Female Cat

Total Cat = 8 + 10 = 18

Step 3: Verify the grand total

Total = Total Dog + Total Cat = 27 + 18 = 45 ✓

Female Cats = 10, Total Cats = 18
Final answer:
DogCatTotal
Male12820
Female151025
Total271845
Applied rules:

• Row sums: Each row must equal its total

• Column sums: Each column must equal its total

• Grand total: Row total sum = Column total sum

2 Finding Relative Frequencies
Exercise 2
Find the relative frequency of male dog owners from the completed table above.
Definition:

Relative Frequency: The proportion of a category compared to the total, expressed as a decimal or percentage

Target Value
Male Dog Owners = 12
Total
Total People = 45
Relative Freq
12/45 = 0.267
Step 1: Identify the value of interest

Male dog owners = 12 people

Step 2: Identify the total

Total surveyed = 45 people

Step 3: Calculate relative frequency

Relative frequency = Part/Total = 12/45 = 0.267 or 26.7%

Relative frequency = 0.267 (26.7%)
Final answer:

The relative frequency of male dog owners is 0.267 or 26.7%

Applied rules:

• Formula: Relative frequency = (specific value)/(total)

• Range: Between 0 and 1 (or 0% and 100%)

• Interpretation: Represents proportion of total

3 Conditional Probability
Exercise 3
Given the completed table, what is the probability that a person owns a cat given they are female?
Definition:

Conditional Probability: The probability of an event occurring given that another event has occurred

Given
Female = 25 people
Condition
Owns Cat = 10 people
Probability
10/25 = 0.4
Step 1: Identify the condition

We're looking only at females (25 people)

Step 2: Find the favorable outcome

Among females, 10 own cats

Step 3: Calculate conditional probability

P(Cat|Female) = Females with cats / Total females = 10/25 = 0.4

P(Cat|Female) = 0.4 (40%)
Final answer:

The probability that a female owns a cat is 0.4 or 40%

Applied rules:

• Formula: P(A|B) = P(A and B) / P(B)

• Condition: Focus only on the given group

• Range: Between 0 and 1 (or 0% and 100%)

Two-Way Tables Fundamentals
Relative\ Frequency = \(\frac{Part}{Total}\)
Relative Frequency Formula
Joint Frequency
Intersection cells
Individual cell values
Marginal Frequency
Row/column totals
Overall category totals
Conditional Prob
P(A|B) = P(A∩B)/P(B)
Probability given condition
Key definitions:

Two-Way Table: A table organizing data by two categorical variables

Joint Frequency: The count in each individual cell

Marginal Frequency: The totals for each row and column

Relative Frequency: Proportion of a value compared to the total

Conditional Probability: Probability of an event given another event occurred

Independence: When knowledge of one variable doesn't affect the probability of another

Two-Way Table Analysis Process:
  1. Understand structure: Identify row and column categories
  2. Find joint frequencies: Values in each cell
  3. Calculate marginal frequencies: Row and column totals
  4. Compute relative frequencies: Proportions of each value
  5. Determine conditional probabilities: Probabilities given conditions
  6. Look for associations: Patterns between variables
Tip 1: Always verify that row totals equal column totals (grand total).
Tip 2: Relative frequencies help compare groups of different sizes.
Tip 3: Conditional probability focuses on a specific subgroup.
Tip 4: Look for patterns that suggest association between variables.
Common errors: Confusing joint and marginal frequencies, forgetting to check totals.
Exam preparation: Practice completing tables, calculating probabilities, interpreting results.
Solution: Exercises 4 to 5
4 Finding Associations
Exercise 4
A survey of 100 students shows: 30 male students like sports, 20 don't; 25 female students like sports, 25 don't. Is there an association between gender and sports preference?
Definition:

Association: When knowing one variable affects the probability of another variable

Completed Table
See below
Male Sports Prob
30/50 = 0.6
Female Sports Prob
25/50 = 0.5
Step 1: Organize the data into a table
Likes SportsDoesn't LikeTotal
Male302050
Female252550
Total5545100
Step 2: Calculate conditional probabilities

P(Likes Sports | Male) = 30/50 = 0.6

P(Likes Sports | Female) = 25/50 = 0.5

Step 3: Compare probabilities

Since 0.6 ≠ 0.5, there appears to be an association between gender and sports preference

Association exists: Males more likely to like sports
Final answer:

Yes, there is an association. Males (60%) are more likely to like sports than females (50%).

Applied rules:

• Association detection: Compare conditional probabilities

• Independence: If probabilities are equal, variables are independent

• Pattern recognition: Different probabilities suggest association

5 Complex Calculations
Exercise 5
From the completed table in Exercise 4, find: a) The relative frequency of students who like sports, b) The probability that a student is male given they like sports.
Definition:

Multi-step Analysis: Performing multiple calculations using the same table

Table Reference
55 like sports out of 100
Rel. Freq.
55/100 = 0.55
Cond. Prob.
30/55 ≈ 0.545
Step 1: Find relative frequency of students who like sports

Students who like sports = 55, Total students = 100

Relative frequency = 55/100 = 0.55 or 55%

Step 2: Find probability that student is male given they like sports

This is conditional probability: P(Male | Likes Sports)

Students who are male AND like sports = 30

Students who like sports = 55

P(Male | Likes Sports) = 30/55 ≈ 0.545 or 54.5%

Step 3: Interpret the results

55% of all students like sports, and among those who like sports, 54.5% are male

Rel. Freq. = 0.55, Cond. Prob. ≈ 0.545
Final answer:

a) The relative frequency of students who like sports is 0.55 (55%)

b) The probability that a student is male given they like sports is approximately 0.545 (54.5%)

Applied rules:

• Relative frequency: Part divided by total

• Conditional probability: Focus on the given condition

• Formula: P(A|B) = P(A and B) / P(B)

Two-Way Tables Analysis Summary
P(A|B) = \(\frac{P(A \cap B)}{P(B)}\)
Conditional Probability Formula
Key definitions:

Two-Way Table: Displays frequency data for two categorical variables

Joint Frequency: Individual cell counts representing intersection of categories

Marginal Frequency: Row and column totals showing overall category frequencies

Relative Frequency: Proportion of a value compared to the total (between 0 and 1)

Conditional Probability: Probability of an event given that another event occurred

Association: When variables are related (knowledge of one affects probability of the other)

Independence: When variables are unrelated (probability of one doesn't depend on the other)

Complete Two-Way Table Analysis:
  1. Structure identification: Determine row and column categories
  2. Value organization: Fill in all known and missing values
  3. Frequency calculation: Find joint, marginal, and relative frequencies
  4. Probability computation: Calculate conditional probabilities
  5. Association detection: Compare conditional probabilities
  6. Interpretation: Explain what the data reveals
Tip 1: Always verify that row totals equal column totals (check for accuracy).
Tip 2: Relative frequencies allow comparison across different-sized groups.
Tip 3: For conditional probability, focus only on the given condition group.
Tip 4: Look for patterns in relative frequencies to identify associations.
Applications: Used in surveys, research, market analysis, and scientific studies.
Limitations: Only works with categorical variables; doesn't prove causation.
Essential Formulas:

• Relative frequency: (specific value) / (total)

• Conditional probability: P(A|B) = P(A and B) / P(B)

• Grand total verification: Sum of row totals = Sum of column totals

• Association detection: Compare P(A|B) with P(A|not B)

Questions & Answers

Question: What's the difference between joint frequency and marginal frequency in a two-way table?

Answer: These terms refer to different locations in a two-way table:

  • Joint frequency: The numbers inside the main body of the table, representing the intersection of two categories (e.g., "male AND likes sports")
  • Marginal frequency: The totals along the margins (edges) of the table, representing overall totals for each category separately

In a table with gender (male/female) and sports preference (yes/no):
Joint frequencies: Male who like sports, Male who don't, Female who like sports, Female who don't
Marginal frequencies: Total males, Total females, Total who like sports, Total who don't

Joint frequencies show the relationship between both variables, while marginal frequencies show each variable independently.

Question: How do I know if there's an association between variables in a two-way table?

Answer: To detect association, compare conditional probabilities:

  1. Calculate conditional probabilities: Find P(outcome|category1), P(outcome|category2), etc.
  2. Compare values: If these probabilities are significantly different, there's likely an association
  3. Look for patterns: Similar probabilities suggest independence, different ones suggest association

For example, if P(likes sports|male) = 0.6 and P(likes sports|female) = 0.3, there's an association because gender affects the probability of liking sports.

If both conditional probabilities were around 0.45, it would suggest little to no association between gender and sports preference.

Question: What's the difference between P(A|B) and P(A and B)? They seem similar but I'm confused.

Answer: These are different concepts with different denominators:

  • P(A and B): Probability that both A and B occur, denominator is total population
  • P(A|B): Probability of A given that B has already occurred, denominator is only those with B

Using our sports example: If 30 out of 100 students are male AND like sports:
P(male AND likes sports) = 30/100 = 0.30
P(male | likes sports) = 30/55 ≈ 0.545 (where 55 is total who like sports)

The key difference: P(A|B) restricts the sample space to only those who satisfy condition B, while P(A and B) considers the entire population.

Question: How do I convert relative frequencies back to actual counts if I need them?

Answer: To convert relative frequency back to actual count, multiply by the total:

Formula: Actual count = Relative frequency × Total

For example, if the relative frequency of male dog owners is 0.267 and the total surveyed is 45:
Actual count = 0.267 × 45 ≈ 12 people

This works because relative frequency was calculated as (count/total), so reversing it gives us back the original count.

Note: You need to know the total to convert back to actual counts. If you only have relative frequencies without the total, you cannot determine the original counts.

Question: Can two-way tables have more than 2 categories for each variable? What if I have 3 types of pets and 3 age groups?

Answer: Yes, absolutely! The "two-way" refers to having two variables, not two categories:

  • Two variables: Pet type and age group (this is still "two-way")
  • Multiple categories: You can have 3+ categories for each variable

Your example would create a 3×3 table (3 pet types × 3 age groups = 9 cells of data plus marginal totals).

The same principles apply: fill in joint frequencies, calculate marginal totals, compute relative frequencies and conditional probabilities. The only difference is having more cells to manage.

Two-way tables can accommodate any number of categories for each variable, making them very versatile for organizing categorical data.