Year 10 Two-Way Tables Worksheets & Practice

Worked examples

Year 10 two-way tables: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

1. Example 1 · Completing a two-way table

   A survey of 80 students recorded whether each student plays a sport (Yes or No) and their year level (Year 9 or Year 10). The partially completed two-way table is shown below.

   	Plays sport	Does not play sport	Total
   Year 9	27	13	40
   Year 10	18	?	40
   Total	?	?	80

   Find the three missing values.

   Show worked solution

   Key idea: Every row total and column total must add up to the grand total of 80.

   1. Find the missing Year 10 frequency for "Does not play sport":
      $40 - 18 = 22$
   2. Find the column total for "Plays sport":
      $27 + 18 = 45$
   3. Find the column total for "Does not play sport":
      $13 + 22 = 35$

   Check: $45 + 35 = 80$ ✓

   Answer: The missing values are 22, 45 and 35.

2. Example 2 · Calculating joint, marginal and conditional frequencies

   A group of 120 adults were surveyed about their preferred type of exercise (Gym or Outdoors) and whether they own a pet (Yes or No). The completed two-way table is below.

   	Gym	Outdoors	Total
   Owns a pet	24	52	76
   Does not own a pet	30	14	44
   Total	54	66	120

   (a) State the joint frequency of adults who prefer Outdoors and own a pet. (b) State the marginal frequency of adults who prefer Gym. (c) Calculate the conditional frequency: of those who prefer Outdoors, what percentage own a pet? Round to one decimal place.

   Show worked solution

   Key idea: A joint frequency is a single cell count; a marginal frequency is a row or column total; a conditional frequency restricts attention to one row or column.

   1. (a) Read the cell where "Outdoors" and "Owns a pet" meet:
      $\text{Joint frequency} = 52$
   2. (b) Read the column total for "Gym":
      $\text{Marginal frequency} = 54$
   3. (c) The total who prefer Outdoors is 66. Of these, 52 own a pet:
      $\frac{52}{66} \times 100 \approx 78.8\%$

   Answer: (a) 52, (b) 54, (c) approximately $78.8\%$

3. Example 3 · Identifying a relationship between two categorical variables

   A school canteen recorded the lunch choice (Hot meal or Sandwich) and the year level (Junior: Years 7 and 8, or Senior: Years 9 and 10) for 200 students on one day.

   	Hot meal	Sandwich	Total
   Junior	72	28	100
   Senior	38	62	100
   Total	110	90	200

   Calculate the percentage of Juniors who chose a hot meal and the percentage of Seniors who chose a hot meal. Use these values to describe whether there appears to be a relationship between year level and lunch choice.

   Show worked solution

   Key idea: To compare groups fairly, convert each row to percentages of its own row total, then compare the percentages across rows.

   1. Percentage of Juniors who chose a hot meal:
      $\frac{72}{100} \times 100 = 72\%$
   2. Percentage of Seniors who chose a hot meal:
      $\frac{38}{100} \times 100 = 38\%$
   3. Compare the two conditional percentages: $72\%$ of Juniors chose a hot meal compared with only $38\%$ of Seniors. The percentages are very different, so the two variables are not distributed in the same way across year levels.
   4. Describe the relationship: Junior students are much more likely to choose a hot meal than Senior students. This suggests there is a relationship between year level and lunch choice, with Juniors favouring hot meals and Seniors favouring sandwiches.

   Answer: Juniors: $72\%$ chose a hot meal; Seniors: $38\%$ chose a hot meal. The large difference suggests there is a relationship between year level and lunch choice.

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