Practice Problems

Problems:

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Problem 1

Explanation: The mode is like the "most popular" answer in a survey. It's the number that shows up the most often. In this table, we'll find out which number of pets is most common by looking at how many households have each number of pets.

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Problem 2

Explanation: The mean is like sharing things out evenly. Imagine everyone reads the same number of books; the mean tells us how many books that would be. To find it, we look at how many books students read and how many students there are.

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Problem 3

Explanation: The median is the "middle" score. Think of it as the middle point if you lined up all the scores in order from smallest to largest. It's like finding the middle person in a line.

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Problem 4

Explanation: The mean age tells us the average age of everyone in the marathon. It's like figuring out how old everyone would be if they were all the same age.

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Problem 5

Explanation: The median shows us the middle value of hours studied. It's like finding the middle student in a group if they all studied for different amounts of time.

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Problem 6

Explanation: The mode is the "most common" weight of the boxes. It's like finding out which weight shows up the most often.

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Solutions:

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Problem 1

Solution:

1. Step 1: Identify the highest frequency in the "Number of Households" row.

• Why? The mode is the value that appears most frequently, so we look for the largest number in the "Number of Households" row.
• The highest frequency is 10.

2. Step 2: Find the corresponding value in the "Number of Pets" row.

• Why? The mode is the value linked to the highest frequency.
• The value corresponding to the highest frequency (10) is 2. Answer: The mode is 2, meaning most households have 2 pets.

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Problem 2

Solution:

3. Step 1: Multiply each number of books by the number of students.

• Why? This step helps us calculate the total number of books read by the students.
• (1 × 3 = 3)
• (2 × 5 = 10)
• (3 × 8 = 24)
• (4 × 4 = 16)
• (5 × 2 = 10)

4. Step 2: Add up all these results.

• Why? This gives us the total number of books read by all the students.
• (3 + 10 + 24 + 16 + 10 = 63)

5. Step 3: Add up the total number of students.

• Why? We need to know the total number of students to divide the total number of books by.
• (3 + 5 + 8 + 4 + 2 = 22)

6. Step 4: Divide the total number of books read by the total number of students.

• Why? Dividing gives us the mean, or the average number of books read per student.
• (63 ÷ 22 ≈ 2.86) Answer: The mean number of books read is approximately 2.86.

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Problem 3

Solution:

7. Step 1: Add up the total frequency (total number of matches).

• Why? We need to know the total number of matches to find the middle value.
• (2 + 4 + 7 + 3 + 1 = 17)

8. Step 2: Find the middle value.

• Why? The median is the middle value, so if there are 17 matches, the median is the 9th value.

9. Step 3: Determine which interval contains the 9th value by adding up the cumulative frequencies:

• Why? The median lies within the interval that contains the middle value.
• First 2 matches: 0 goals
• Next 4 matches: 1 goal (cumulative frequency = 6)
• Next 7 matches: 2 goals (cumulative frequency = 13)
• The 9th value is within the 2-goal interval. Answer: The median number of goals scored is 2.

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Problem 4

Solution:

10. Step 1: Find the mid-interval value for each age group.

• Why? Since we're working with intervals, we use the mid-interval value as a representative for the entire group.
• 10-20: (10 + 20)/2 = 15
• 20-30: (20 + 30)/2 = 25
• 30-40: (30 + 40)/2 = 35
• 40-50: (40 + 50)/2 = 45
• 50-60: (50 + 60)/2 = 55

11. Step 2: Multiply each mid-interval value by its frequency.

• Why? This step gives us the total contribution of each age group to the overall mean.
• (15 × 6 = 90)
• (25 × 10 = 250)
• (35 × 12 = 420)
• (45 × 8 = 360)
• (55 × 4 = 220)

12. Step 3: Add up all these results.

• Why? This gives us the total sum of all ages (in terms of mid-interval values).
• (90 + 250 + 420 + 360 + 220 = 1340)

13. Step 4: Add up the total frequency.

• Why? We divide the total sum by the total number of participants to find the mean.
• (6 + 10 + 12 + 8 + 4 = 40)

14. Step 5: Divide the total by the total frequency.

• Why? Dividing gives us the mean, or the average age of the participants.
• (1340 ÷ 40 = 33.5) Answer: The mean age of the participants is 33.5 years.

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Problem 5

Solution:

15. Step 1: Add up the total frequency (total number of students).

• Why? We need to know how many students there are to find the middle value.
• (3 + 8 + 12 + 5 + 2 = 30)

16. Step 2: Find the middle value.

• Why? The median is the middle value, so for 30 students, the median is between the 15th and 16th values.

17. Step 3: Determine which interval contains the median by adding up the cumulative frequencies:

• Why? The median lies within the interval that contains the middle values.
• First 3 values: 0-5 hours
• Next 8 values: 5-10 hours (cumulative frequency = 11)
• Next 12 values: 10-15 hours (cumulative frequency = 23)
• The 15th and 16th values fall in the 10-15 hours interval.

18. Step 4: Use the mid-interval value of that interval (10-15 hours).

• Why? The median is represented by the mid-interval value of the group where the middle values fall.
• Mid-interval value: (10 + 15)/2 = 12.5 Answer: The median number of hours studied is 12.5 hours.

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Problem 6

Solution:

19. Step 1: Identify the interval with the highest frequency.

• Why? The mode is the value that appears most frequently, so we look for the interval with the highest frequency.
• The highest frequency is 15, which occurs in the 20-30 kg interval.

20. Step 2: The mode is the mid-interval value of that interval.

• Why? The mode is represented by the mid-interval value of the group with the highest frequency.
• Mid-interval value for 20-30 kg: (20 + 30)/2 = 25 Answer: The mode is 25 kg.

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