Q1: A two-way table shows the relationship between two categorical variables. What is another name for a two-way table? (a) Frequency distribution (b) Contingency table (c) Histogram (d) Box plot
Solution:
Ans: (b) Explanation: A contingency table is another name for a two-way table. Both terms refer to a table that displays the frequency distribution of two categorical variables. A frequency distribution shows data for one variable, while histograms and box plots are graphical representations, not tables.
Q2: In a two-way table, the numbers inside the table (not in the margins) are called: (a) Marginal frequencies (b) Joint frequencies (c) Relative frequencies (d) Conditional frequencies
Solution:
Ans: (b) Explanation: The numbers inside a two-way table are called joint frequencies because they represent the frequency of two categorical variables occurring together. Marginal frequencies are the totals in the margins, relative frequencies are proportions, and conditional frequencies are frequencies given a specific condition.
Q3: A survey of 100 students shows that 60 like pizza and 40 like burgers. If 25 students like both, how many students like only pizza? (a) 25 (b) 35 (c) 60 (d) 85
Solution:
Ans: (b) Explanation: To find students who like only pizza, subtract those who like both from the total who like pizza: \(60 - 25 = 35\). Option (a) is the number who like both, (c) is the total who like pizza, and (d) is the sum of those who like pizza or burgers.
Q4: In a two-way table with 150 total observations, if one cell has a joint frequency of 45, what is the relative frequency of that cell? (a) 0.30 (b) 0.45 (c) 3.33 (d) 45
Solution:
Ans: (a) Explanation:Relative frequency is calculated by dividing the joint frequency by the total: \(\frac{45}{150} = 0.30\). Option (b) represents the joint frequency as a decimal, (c) would result from incorrect division, and (d) is the joint frequency itself.
Q5: A two-way table shows data on gender (Male, Female) and sport preference (Soccer, Basketball). If there are 30 males who prefer soccer out of 50 total males, what is the conditional relative frequency of males preferring soccer given they are male? (a) 0.30 (b) 0.50 (c) 0.60 (d) 30
Solution:
Ans: (c) Explanation: The conditional relative frequency is calculated by dividing the specific category by its row or column total: \(\frac{30}{50} = 0.60\). This represents the proportion of males who prefer soccer. Option (a) would be incorrect division, (b) represents the total males as a decimal, and (d) is the raw frequency.
Q6: Which of the following statements about marginal frequencies is true? (a) They are always less than joint frequencies (b) They are found in the interior cells of the table (c) They represent the totals for each category of a variable (d) They show the relationship between two variables
Solution:
Ans: (c) Explanation:Marginal frequencies represent the totals for each category of a variable and are found in the margins (edges) of the table. Option (a) is incorrect because marginal frequencies are sums and can be larger, (b) is incorrect because they're in the margins, and (d) describes joint frequencies.
Q7: A two-way table shows 80 students categorized by grade level (Freshman, Sophomore) and club membership (Yes, No). If 35 are freshmen, 45 are sophomores, and 50 are club members, how many sophomores are club members if 28 freshmen are club members? (a) 7 (b) 22 (c) 23 (d) 50
Solution:
Ans: (b) Explanation: Since 50 total students are club members and 28 freshmen are club members, the number of sophomore club members is: \(50 - 28 = 22\). Option (a) represents the difference in non-members, (c) is close but incorrect, and (d) is the total club members.
Q8: In a two-way relative frequency table, all the relative frequencies in the entire table must sum to: (a) 0 (b) 1 (c) 10 (d) 100
Solution:
Ans: (b) Explanation: In a two-way relative frequency table, all relative frequencies must sum to 1 (or 100%) because they represent proportions of the whole. Option (a) is incorrect, (c) would only occur with percentages expressed as decimals incorrectly, and (d) applies to percentages, not decimal relative frequencies.
## Section B: Fill in the Blanks Q9: The sum of all joint frequencies in a two-way table equals the __________ frequency.
Solution:
Ans: total Explanation: The sum of all joint frequencies in a two-way table equals the total frequency, which represents the total number of observations in the dataset.
Q10: A __________ relative frequency is found by dividing a joint frequency by the grand total.
Solution:
Ans: joint Explanation: A joint relative frequency is calculated by dividing a joint frequency (the frequency of two categories occurring together) by the grand total of all observations.
Q11: When calculating a conditional relative frequency, you divide the joint frequency by the appropriate __________ frequency.
Solution:
Ans: marginal Explanation: To find a conditional relative frequency, divide the joint frequency by the appropriate marginal frequency (the row or column total), which gives the proportion within that specific category.
Q12: In a two-way table, the frequencies in the margins are called __________ frequencies.
Solution:
Ans: marginal Explanation: The frequencies found in the margins (edges) of a two-way table are called marginal frequencies, representing the totals for each category of one variable.
Q13: A two-way table is used to organize data for __________ categorical variables.
Solution:
Ans: two Explanation: A two-way table is specifically designed to organize and display data for two categorical variables simultaneously, showing their relationship.
Q14: The relative frequency of a category is expressed as a decimal between 0 and __________ or as a percentage between 0% and 100%.
Solution:
Ans: 1 Explanation:Relative frequencies are proportions and must fall between 0 and 1 when expressed as decimals, or between 0% and 100% when expressed as percentages.
## Section C: Word Problems Q15: A high school surveyed 200 students about their preference for online or in-person classes and whether they are in Grade 9 or Grade 10. The results showed that 85 students prefer online classes, 60 of whom are in Grade 9. If there are 110 Grade 9 students total, how many Grade 9 students prefer in-person classes?
Solution:
Ans: Step 1: Total Grade 9 students = 110 Step 2: Grade 9 students who prefer online = 60 Step 3: Grade 9 students who prefer in-person = \(110 - 60 = 50\) Final Answer: 50 students
Q16: A store tracks customer purchases by payment method (Cash or Card) and time of day (Morning or Afternoon). The two-way table shows: 45 customers paid by cash in the morning, 30 paid by cash in the afternoon, 55 paid by card in the morning, and 70 paid by card in the afternoon. What is the marginal frequency for afternoon purchases?
Q17: In a survey of 150 teenagers about their favorite social media platform (Instagram or TikTok) and gender (Male or Female), 65 males were surveyed. If 40 males prefer Instagram and 55 females prefer TikTok, how many females prefer Instagram?
Solution:
Ans: Step 1: Total teenagers = 150 Step 2: Total males = 65, so total females = \(150 - 65 = 85\) Step 3: Females who prefer TikTok = 55 Step 4: Females who prefer Instagram = \(85 - 55 = 30\) Final Answer: 30 females
Q18: A school cafeteria recorded lunch choices (Salad or Sandwich) for 240 students. The two-way table shows that 90 students chose salad. If the joint relative frequency of students choosing sandwiches is 0.625, how many students chose sandwiches?
Solution:
Ans: Step 1: Total students = 240 Step 2: Joint relative frequency for sandwiches = 0.625 Step 3: Number of students who chose sandwiches = \(240 \times 0.625 = 150\) Final Answer: 150 students
Q19: A fitness center surveyed 180 members about their workout preference (Cardio or Weights) and membership type (Monthly or Annual). The data shows 70 monthly members, of which 42 prefer cardio. If 65 total members prefer weights, find the conditional relative frequency of annual members preferring weights, given they have annual membership. Round your answer to two decimal places.
Solution:
Ans: Step 1: Total members = 180, Monthly members = 70 Step 2: Annual members = \(180 - 70 = 110\) Step 3: Monthly members who prefer cardio = 42 Step 4: Monthly members who prefer weights = \(70 - 42 = 28\) Step 5: Total who prefer weights = 65 Step 6: Annual members who prefer weights = \(65 - 28 = 37\) Step 7: Conditional relative frequency = \(\frac{37}{110} = 0.336... \approx 0.34\) Final Answer: 0.34
Q20: A library surveyed 300 visitors about their reading preference (Fiction or Non-fiction) and age group (Youth or Adult). The two-way table shows that 120 visitors are youth, 100 prefer fiction, and 45 youth prefer fiction. Complete the two-way table and determine how many adults prefer non-fiction.
Solution:
Ans: Step 1: Total visitors = 300, Youth = 120 Step 2: Adults = \(300 - 120 = 180\) Step 3: Total who prefer fiction = 100, Youth who prefer fiction = 45 Step 4: Adults who prefer fiction = \(100 - 45 = 55\) Step 5: Adults who prefer non-fiction = \(180 - 55 = 125\) Final Answer: 125 adults
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