# Assistant: Apologies, but I notice there's a formatting error in my instructions. Let me provide the complete worksheet following all the rules exactly:
Section A: Multiple Choice Questions
Q1: A survey was conducted among 100 students asking whether they prefer basketball or soccer. If 60 students prefer basketball and 40 prefer soccer, what type of data is being collected? (a) Numerical data only (b) Categorical data only (c) Both numerical and categorical data (d) Neither numerical nor categorical data
Solution:
Ans: (b) Explanation: The data collected is categorical because it classifies students into categories (basketball or soccer) rather than measuring numerical values. The counts are numerical summaries of categorical data, but the variable itself is categorical.
Q2: In a two-way table showing gender (male, female) and favorite subject (Math, English, Science), if there are 3 rows and 3 columns visible, what do the extra row and column represent? (a) Data entry errors (b) Total counts for each category (c) Additional variables (d) Duplicate data
Solution:
Ans: (b) Explanation: In a two-way table, the extra row and column represent marginal totals or total counts for each category. These totals help summarize the data by showing row totals, column totals, and the grand total.
Q3: A two-way table shows 80 students classified by grade level (9th or 10th) and whether they have a part-time job (yes or no). If 30 ninth graders have jobs and 50 students total are in 9th grade, how many 9th graders do NOT have jobs? (a) 20 (b) 30 (c) 50 (d) 80
Solution:
Ans: (a) Explanation: Since there are 50 total ninth graders and 30 have jobs, the number without jobs is \(50 - 30 = 20\). This represents a cell value within the two-way table found by subtracting from the marginal total.
Q4: In a two-way frequency table, what does a cell in the body of the table represent? (a) The total for a row (b) The total for a column (c) The count of observations in both categories (d) The grand total of all observations
Solution:
Ans: (c) Explanation: Each cell in the body of a two-way table represents the count (frequency) of observations that fall into both the row category and the column category simultaneously. This is called a joint frequency.
Q5: A two-way table shows 120 students by eye color (brown, blue, green) and hair color (blonde, brunette). If 45 students have brown eyes, what type of total is 45? (a) Joint frequency (b) Relative frequency (c) Marginal frequency (d) Conditional frequency
Solution:
Ans: (c) Explanation: The value 45 represents a marginal frequency because it is the total count for one category (brown eyes) across all categories of the other variable (hair color). Marginal frequencies appear in the margins (edges) of the table.
Q6: In a two-way relative frequency table, all entries are expressed as what? (a) Whole numbers (b) Decimals or percentages (c) Fractions only (d) Ratios greater than 1
Solution:
Ans: (b) Explanation: A relative frequency table expresses entries as decimals or percentages representing the proportion of the total. Each value is calculated by dividing the frequency by the total number of observations.
Q7: A two-way table shows 200 students classified by whether they play sports (yes or no) and whether they are in band (yes or no). If 75 students play sports and are in band, and 125 students total play sports, what is the relative frequency of students who play sports AND are in band? (a) 0.375 (b) 0.600 (c) 0.625 (d) 0.750
Solution:
Ans: (a) Explanation: The relative frequency is calculated by dividing the joint frequency by the grand total: \(\frac{75}{200} = 0.375\) or 37.5%. This represents the proportion of all students who both play sports and are in band.
Q8: In analyzing a two-way table, if you want to find the percentage of students who prefer pizza given that they are female, what type of frequency are you calculating? (a) Joint relative frequency (b) Marginal relative frequency (c) Conditional relative frequency (d) Total relative frequency
Solution:
Ans: (c) Explanation: A conditional relative frequency is calculated when finding the percentage within a specific category (given that they are female). It is found by dividing the joint frequency by the marginal total of the condition (total females).
Section B: Fill in the Blanks
Q9: A table that displays data for two categorical variables is called a __________.
Solution:
Ans: two-way table (or two-way frequency table) Explanation: A two-way table organizes data by showing the relationship between two categorical variables in rows and columns.
Q10: In a two-way table, the totals shown at the end of each row and column are called __________ frequencies.
Solution:
Ans: marginal Explanation:Marginal frequencies are the row and column totals that appear in the margins of a two-way table, showing the total count for each category of a single variable.
Q11: The total number of all observations in a two-way table is called the __________ total.
Solution:
Ans: grand Explanation: The grand total is the sum of all observations in the table, typically located in the bottom-right cell where the row and column totals meet.
Q12: A frequency found by dividing a cell frequency by the grand total is called a __________ relative frequency.
Solution:
Ans: joint Explanation: A joint relative frequency represents the proportion of the total that falls into both categories simultaneously, calculated as the cell value divided by the grand total.
Q13: When calculating a conditional relative frequency, you divide the joint frequency by the __________ total of the given condition.
Solution:
Ans: marginal Explanation:Conditional relative frequency is found by dividing the joint frequency by the marginal total of the specified condition, giving the proportion within that specific category.
Q14: In a two-way table, data that can be placed into distinct categories such as "yes" or "no" is called __________ data.
Solution:
Ans: categorical Explanation:Categorical data consists of values that can be sorted into groups or categories rather than measured numerically.
Section C: Word Problems
Q15: A high school surveyed 150 students about whether they have a smartphone and whether they have a tablet. The results showed that 90 students have a smartphone, 60 students have a tablet, and 40 students have both. Create a two-way table and determine how many students have neither device.
Solution:
Ans: Final Answer: 40 students have neither device
Explanation: Students with only smartphone: \(90 - 40 = 50\) Students with only tablet: \(60 - 40 = 20\) Students with both: 40 Students with at least one device: \(50 + 20 + 40 = 110\) Students with neither: \(150 - 110 = 40\)
Q16: A survey of 200 ninth graders asked about their favorite season (Summer or Winter) and whether they prefer indoor or outdoor activities. The two-way table shows that 75 students prefer Summer and outdoor activities, 45 prefer Summer and indoor activities, 30 prefer Winter and outdoor activities, and 50 prefer Winter and indoor activities. What is the marginal frequency for students who prefer outdoor activities?
Solution:
Ans: Final Answer: 105 students prefer outdoor activities
Explanation: Students preferring outdoor activities = Summer/Outdoor + Winter/Outdoor \(75 + 30 = 105\) students
Q17: In a class of 80 students, a two-way table shows enrollment in Math Club and Science Club. If 25 students are in Math Club only, 30 are in Science Club only, and 15 are in both clubs, calculate the joint relative frequency for students in both clubs. Express your answer as a decimal.
Solution:
Ans: Final Answer: 0.1875
Explanation: Joint relative frequency = \(\frac{\text{Students in both clubs}}{\text{Total students}}\) \(\frac{15}{80} = 0.1875\) or 18.75%
Q18: A two-way table categorizes 240 students by grade (9th or 10th) and whether they walk to school (yes or no). The table shows 100 total ninth graders, and 60 ninth graders walk to school. If 80 total students walk to school, how many tenth graders walk to school?
Solution:
Ans: Final Answer: 20 tenth graders walk to school
Explanation: Total students who walk = 80 Ninth graders who walk = 60 Tenth graders who walk = \(80 - 60 = 20\) students
Q19: A store surveyed 300 customers about their purchase of electronics. The two-way table shows that 180 customers bought a laptop, 120 bought headphones, and 75 bought both items. Find the conditional relative frequency of customers who bought headphones given that they bought a laptop. Round to three decimal places.
Solution:
Ans: Final Answer: 0.417
Explanation: Conditional relative frequency = \(\frac{\text{Customers who bought both}}{\text{Customers who bought laptop}}\) \(\frac{75}{180} = 0.41\overline{6} \approx 0.417\)
Q20: A two-way frequency table shows 160 students classified by whether they participate in drama (yes or no) and choir (yes or no). If 50 students participate in drama, 70 participate in choir, and 25 participate in both, what is the marginal relative frequency for students who participate in drama? Express as a percentage.
Solution:
Ans: Final Answer: 31.25%
Explanation: Marginal relative frequency = \(\frac{\text{Students in drama}}{\text{Total students}}\) \(\frac{50}{160} = 0.3125 = 31.25\%\)
The document Worksheet (with Solutions): Two-Way Tables is a part of the Grade 9 Course Integrated Math 1.
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