Q1: A survey asked 100 students to choose their favorite type of music from the following options: Pop, Rock, Hip-Hop, and Country. What type of variable is "favorite type of music"? (a) Quantitative (b) Categorical (c) Continuous (d) Numerical
Solution:
Ans: (b) Explanation:Favorite type of music is a categorical variable because it represents categories or groups (Pop, Rock, Hip-Hop, Country) rather than numerical values. Quantitative and numerical variables involve measurements or counts, while continuous variables can take any value within a range.
Q2: A frequency table shows that 25 out of 80 students prefer chocolate ice cream. What is the relative frequency of students who prefer chocolate ice cream? (a) 0.25 (b) 0.3125 (c) 25 (d) 3.2
Solution:
Ans: (b) Explanation:Relative frequency is calculated by dividing the frequency by the total number of observations. Calculation: \(\frac{25}{80} = 0.3125\) Option (a) represents \(\frac{25}{100}\), option (c) is the frequency itself, and option (d) is an incorrect calculation.
Q3: In a bar graph displaying favorite sports of students, which axis typically shows the categories? (a) The vertical axis (y-axis) (b) The horizontal axis (x-axis) (c) Both axes equally (d) Neither axis
Solution:
Ans: (b) Explanation: In a bar graph for categorical data, the categories are typically displayed on the horizontal axis (x-axis), while the frequency or count is shown on the vertical axis (y-axis). This is the standard convention, though vertical bar graphs can also be used.
Q4: A pie chart shows that the central angle for the "Red" category is 90°. If the total number of observations is 200, how many observations are in the "Red" category? (a) 25 (b) 45 (c) 50 (d) 90
Solution:
Ans: (c) Explanation: The central angle in a pie chart represents the proportion of the category. Calculation: \(\frac{90°}{360°} = \frac{1}{4}\) Number of observations: \(\frac{1}{4} \times 200 = 50\) Option (a) would result from \(\frac{90}{360} \times 100\), option (b) is incorrect, and option (d) confuses the angle with the count.
Q5: Which of the following is NOT an appropriate way to display categorical data? (a) Bar graph (b) Pie chart (c) Histogram (d) Frequency table
Solution:
Ans: (c) Explanation: A histogram is used to display quantitative data, not categorical data. Bar graphs, pie charts, and frequency tables are all appropriate methods for displaying categorical data. The key difference is that histograms show continuous data with no gaps between bars.
Q6: A frequency table shows the following data for favorite colors: Blue (15), Red (20), Green (10), Yellow (5). What percentage of people chose Red? (a) 20% (b) 25% (c) 40% (d) 50%
Solution:
Ans: (c) Explanation: First, find the total number of observations: \(15 + 20 + 10 + 5 = 50\) Then calculate the percentage: \(\frac{20}{50} \times 100\% = 40\%\) Option (a) confuses the frequency with the percentage, option (b) would be correct if the total were 80, and option (d) represents half of the total.
Q7: In analyzing categorical data, which measure is most appropriate to identify the most common category? (a) Mean (b) Median (c) Mode (d) Range
Solution:
Ans: (c) Explanation: The mode is the value or category that appears most frequently in a dataset, making it the most appropriate measure for identifying the most common category in categorical data. Mean and median are measures for quantitative data, and range measures spread.
Q8: A two-way frequency table is used to analyze the relationship between two categorical variables. If one variable is "Gender" with 2 categories and another is "Favorite Season" with 4 categories, how many cells (not including totals) will the table have? (a) 2 (b) 4 (c) 6 (d) 8
Solution:
Ans: (d) Explanation: A two-way frequency table has cells equal to the product of the number of categories in each variable. Calculation: \(2 \times 4 = 8\) cells Options (a), (b), and (c) represent incomplete calculations of the table structure.
Section B: Fill in the Blanks
Q9: A __________ is a table that shows the number of observations in each category of a categorical variable.
Solution:
Ans: frequency table Explanation: A frequency table organizes categorical data by listing each category along with its count or frequency, making it easy to see how observations are distributed across categories.
Q10: The __________ frequency is the ratio of the frequency of a category to the total number of observations, often expressed as a decimal or percentage.
Solution:
Ans: relative Explanation:Relative frequency provides the proportion of observations in each category compared to the total, calculated as frequency divided by total number of observations.
Q11: A __________ chart is a circular graph divided into sectors, where each sector represents a category and its size is proportional to its frequency.
Solution:
Ans: pie Explanation: A pie chart uses a circle divided into slices to show the relative sizes of categories, with each slice's central angle proportional to the category's frequency.
Q12: In a bar graph for categorical data, the bars should not __________ each other because the categories are distinct.
Solution:
Ans: touch Explanation: In bar graphs for categorical data, bars are separated by gaps to emphasize that the categories are distinct and not continuous, unlike histograms where bars touch.
Q13: A __________ distribution shows that all categories have approximately the same frequency.
Solution:
Ans: uniform Explanation: A uniform distribution occurs when observations are evenly spread across all categories, meaning each category has roughly equal frequency.
Q14: The sum of all relative frequencies in a frequency distribution must equal __________.
Solution:
Ans: 1 (or 100%) Explanation: Since relative frequencies represent proportions of the whole, their sum must equal 1 (or 100% if expressed as percentages), accounting for all observations in the dataset.
Section C: Word Problems
Q15: A high school surveyed 150 students about their preferred method of transportation to school. The results were: Bus (60 students), Car (45 students), Bike (30 students), and Walk (15 students). Create a frequency table and calculate the relative frequency for each category.
Solution:
Ans: Frequency Table: Bus: 60, Relative Frequency = \(\frac{60}{150} = 0.40\) or 40% Car: 45, Relative Frequency = \(\frac{45}{150} = 0.30\) or 30% Bike: 30, Relative Frequency = \(\frac{30}{150} = 0.20\) or 20% Walk: 15, Relative Frequency = \(\frac{15}{150} = 0.10\) or 10% Final Answer: Bus (0.40), Car (0.30), Bike (0.20), Walk (0.10)
Q16: A pet store recorded the types of pets sold in one month: Dogs (40), Cats (35), Fish (20), Birds (15), and Hamsters (10). If you were to create a pie chart, what would be the central angle for the "Cats" category?
Solution:
Ans: Step 1: Find total number of pets: \(40 + 35 + 20 + 15 + 10 = 120\) Step 2: Calculate the proportion for Cats: \(\frac{35}{120}\) Step 3: Calculate central angle: \(\frac{35}{120} \times 360° = 105°\) Final Answer: 105°
Q17: A movie theater tracked the genre preferences of 200 moviegoers. The bar graph showed: Action (70), Comedy (55), Drama (40), Horror (25), and Science Fiction (10). What is the mode of this distribution, and what percentage of moviegoers preferred this genre?
Solution:
Ans: Step 1: Identify the mode (most common category): Action with 70 moviegoers Step 2: Calculate percentage: \(\frac{70}{200} \times 100\% = 35\%\) Final Answer: Mode is Action; 35% of moviegoers preferred Action
Q18: A teacher surveyed 80 students about their favorite subject. The relative frequencies were: Math (0.25), Science (0.30), English (0.20), History (0.15), and Art (0.10). How many students chose Science as their favorite subject?
Solution:
Ans: Step 1: Use the formula: Frequency = Relative Frequency × Total Step 2: Calculate: \(0.30 \times 80 = 24\) Final Answer: 24 students chose Science
Q19: A survey asked 120 people to name their favorite season. Spring was chosen by 30 people, Summer by 45 people, Fall by 25 people, and Winter by 20 people. Construct a bar graph on paper and determine which season has the highest bar. Then calculate the difference in frequency between the most and least popular seasons.
Solution:
Ans: Step 1: Identify the season with highest frequency: Summer (45) Step 2: Identify the season with lowest frequency: Winter (20) Step 3: Calculate difference: \(45 - 20 = 25\) Final Answer: Summer has the highest bar; the difference between most and least popular is 25 people
Q20: A clothing store categorized customer purchases over a week: Shirts (85), Pants (60), Shoes (45), Accessories (30), and Jackets (20). The store manager wants to create a visual display showing what percentage each category represents. Calculate the percentage for each category and identify which category represents exactly one-quarter of all purchases.
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