Solved Exercises on Statistical Questions in Grade 7

Master statistical questions: identifying statistical vs non-statistical questions, variability, and data collection methods through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Identifying Statistical Questions
Exercise 1
Determine whether each question is statistical or non-statistical. Explain your reasoning.
a) How old is Sarah?
b) How old are the students in my school?
c) What is the capital of France?
Definition:

Statistical Question: A question that anticipates variability in the data related to the question and accounts for it in the answers.

Non-Statistical Question: A question that has one specific answer with no variability.

Identification Method:
  1. Read the question carefully
  2. Determine if the question expects multiple possible answers
  3. Ask yourself: "Will the answers vary?"
  4. If yes, it's a statistical question; if no, it's non-statistical
  5. Look for words like "all," "each," or specific individuals (non-statistical)
  6. Look for groups, populations, or ranges (statistical)
Question a
Non-statistical
Question b
Statistical
Question c
Non-statistical
Step 1: Analyze Question a - "How old is Sarah?"

This question asks for one specific answer about one person.

There is no variability expected in the answer.

Step 2: Analyze Question b - "How old are the students in my school?"

This question expects multiple different answers since students have different ages.

There is significant variability in the responses.

Step 3: Analyze Question c - "What is the capital of France?"

This question has one definitive answer (Paris).

No variability is expected in the response.

Step 4: Classify each question

a) Non-statistical - one answer

b) Statistical - multiple possible answers

c) Non-statistical - one answer

a) Non-statistical, b) Statistical, c) Non-statistical
Final Answer:

Question a and c are non-statistical; question b is statistical.

Applied Rules:

• Variability Test: Statistical questions anticipate varied responses

• Single Answer Test: Non-statistical questions have one correct answer

• Population vs Individual: Questions about groups are typically statistical

2 Variability Analysis
Exercise 2
Which of these questions would produce data with the greatest variability? Explain your reasoning.
a) How tall are adult males in the United States?
b) How tall are students in 7th grade at Lincoln Middle School?
c) How tall are members of the basketball team?
Definition:

Variability: The degree to which data points differ from each other and from the mean.

Data Range: The difference between the maximum and minimum values in a dataset.

Option a
High variability
Option b
Medium variability
Option c
Low variability
Step 1: Analyze Option a - Adult males in the US

This group includes men of all ages (18+), from teenagers to elderly.

Age range: 18 to 100+ years, with potential height variation.

Step 2: Analyze Option b - 7th grade students

All subjects are approximately the same age (12-13 years).

More uniform age group, but still growing at different rates.

Step 3: Analyze Option c - Basketball team members

Team members likely selected based on height requirements.

Expected to have similar heights, with limited variation.

Step 4: Compare variability levels

Option a: Wide age range → greatest height variation

Option b: Similar age → moderate variation

Option c: Selected for height → least variation

Option a has the greatest variability
Final Answer:

Question a "How tall are adult males in the United States?" would produce data with the greatest variability because it encompasses the widest age range and most diverse population.

Applied Rules:

• Sample Size Effect: Larger, more diverse samples tend to have higher variability

• Selection Criteria: Groups selected for specific characteristics have lower variability

• Demographics: Age, gender, and other factors affect data spread

3 Population vs Sample
Exercise 3
Identify the population and sample in this scenario: A researcher wants to know the favorite ice cream flavor of all 7th graders in a city. She surveys 200 randomly selected 7th graders from different schools.
Definition:

Population: The entire group of individuals or items that researchers want to study.

Sample: A subset of the population that is actually studied.

Random Sampling: A method where every member of the population has an equal chance of being selected.

Population
All 7th graders in the city
Sample
200 selected students
Sampling Method
Random sampling
Step 1: Identify the research goal

The researcher wants to know about "all 7th graders in a city"

This represents the target population.

Step 2: Identify who was actually surveyed

"200 randomly selected 7th graders from different schools"

This represents the sample.

Step 3: Analyze the sampling method

Students were randomly selected from different schools.

This is random sampling to ensure representation.

Step 4: Evaluate the sample's representativeness

The sample is taken from different schools to represent the entire city.

This increases the likelihood of accurate results.

Population: All 7th graders in the city
Sample: 200 selected students
Final Answer:

Population: All 7th graders in the city. Sample: 200 randomly selected 7th graders from different schools.

Applied Rules:

• Population vs Sample: Population is the entire group, sample is the studied subset

• Representative Sampling: Sample should reflect population characteristics

• Random Selection: Reduces bias in sampling

Rules and methods, laws,...
\(\text{Statistical Question} = \text{Anticipates Variability in Data}\)
Statistical Question Principle
Population vs Sample
Population ⊃ Sample
Sample is subset of population
Variability Measure
Range = Max - Min
Difference between extremes
Data Collection
Random Sampling
Reduces bias
Key Definitions:

Statistical Question: A question that anticipates variability in the data related to the question and accounts for it in the answers

Population: The entire group of individuals or items that researchers want to study

Sample: A subset of the population that is actually studied

Variability: The extent to which data points differ from each other and from the mean

Random Sampling: A method where every member of the population has an equal chance of being selected

Complete Methodology:
  1. Identify the Question Type: Determine if it's statistical or non-statistical
  2. Analyze Expected Responses: Will answers vary or remain constant?
  3. Define Population and Sample: Identify who/what is being studied
  4. Assess Variability: Consider the range and diversity of expected responses
  5. Evaluate Sampling Method: Determine how participants were selected
  6. Draw Conclusions: Based on the statistical nature of the question
Tip 1: Statistical questions ask about groups, not individuals.
Tip 2: Look for words like "average," "typical," "range," or "distribution."
Tip 3: If you expect multiple different answers, it's likely statistical.
Tip 4: Non-statistical questions have one definitive answer.
Tip 5: Larger, more diverse populations typically have higher variability.
Common Errors: Confusing individual vs group questions, misidentifying population/sample, underestimating variability.
Exam Preparation: Practice identifying question types, understand sampling methods, know variability concepts.
Solution: Exercises 4 to 5
4 Survey Design
Exercise 4
Design a statistical question to determine the most popular extracurricular activity among students in your school. Explain why your question is statistical and describe how you would collect the data.
Definition:

Survey Question: A question designed to gather information from a group of people.

Data Collection Method: The process of gathering information systematically.

Statistical Question
What extracurricular activities do students participate in?
Data Collection
Survey method
Population
All students in school
Step 1: Formulate the statistical question

"Which extracurricular activity do students in our school participate in most frequently?"

This question anticipates varied responses from different students.

Step 2: Explain why it's statistical

Different students participate in different activities.

Responses will vary significantly across the student population.

Step 3: Describe data collection method

Create a survey with all possible extracurricular activities.

Distribute to a random sample of students across grades.

Step 4: Identify population and sample

Population: All students in the school

Sample: Randomly selected students from each grade

Step 5: Plan data analysis

Count responses for each activity.

Calculate percentages and identify the most popular activity.

Statistical question: What extracurricular activities do students participate in?
Final Answer:

Statistical question: "Which extracurricular activity do students in our school participate in most frequently?" This is statistical because different students participate in different activities, leading to varied responses. Data would be collected through a survey distributed to a random sample of students.

Applied Rules:

• Statistical Question Requirements: Must anticipate variability in responses

• Representative Sampling: Sample should reflect population characteristics

• Data Collection: Systematic approach ensures reliability

5 Data Interpretation
Exercise 5
A survey asked 100 students "How many hours per week do you spend on homework?" The responses ranged from 2 to 15 hours. Calculate the range and explain what this tells us about the data.
Definition:

Range: The difference between the maximum and minimum values in a dataset.

Data Spread: How much the data points are spread out across the range of values.

Minimum
2 hours
Maximum
15 hours
Range
13 hours
Step 1: Identify minimum and maximum values

Minimum: 2 hours per week

Maximum: 15 hours per week

Step 2: Apply range formula

Range = Maximum - Minimum

Range = 15 - 2 = 13 hours

Step 3: Interpret the range value

A range of 13 hours indicates substantial variability in homework time.

Students spend widely different amounts of time on homework.

Step 4: Analyze implications

The large range suggests diverse study habits among students.

Some students spend minimal time, others spend significantly more.

Step 5: Consider further analysis

Additional measures (mean, median, mode) would provide more insight.

A frequency distribution could show the pattern of responses.

Range = 13 hours, indicating high variability
Final Answer:

The range is 13 hours (15 - 2 = 13). This indicates high variability in the amount of time students spend on homework, ranging from 2 to 15 hours per week.

Applied Rules:

• Range Formula: Range = Maximum - Minimum

• Variability Indicator: Larger range indicates greater spread in data

• Data Interpretation: Range provides initial insight into data distribution

Detailed Summary: Statistical Questions Fundamentals
\(\text{Statistical Question} = \text{Anticipates Variability in Data}\)
Core Principle
Key definitions:

Statistical Question: A question that anticipates variability in the data related to the question and accounts for it in the answers

Non-Statistical Question: A question that has one specific answer with no variability

Population: The entire group of individuals or items that researchers want to study

Sample: A subset of the population that is actually studied

Variability: The extent to which data points differ from each other and from the mean

Range: The difference between the maximum and minimum values in a dataset

Complete methodology:
  1. Question Classification: Determine if the question anticipates variability in responses
  2. Identify Components: Recognize population, sample, and sampling method
  3. Assess Variability: Consider the expected range and diversity of responses
  4. Design Data Collection: Plan systematic approach for gathering information
  5. Analyze Data: Calculate measures like range, mean, median, or mode
  6. Draw Conclusions: Interpret results based on statistical principles
Tip 1: Statistical questions always involve groups or populations, never single individuals.
Tip 2: Ask yourself: "Will different people give different answers?" If yes, it's statistical.
Tip 3: Larger, more diverse groups typically show greater variability in responses.
Tip 4: Random sampling helps ensure representative results.
Tip 5: Range is just one measure of variability; consider other measures for complete analysis.
Common errors: Misidentifying statistical vs non-statistical questions, confusing population with sample, underestimating variability impact.
Exam preparation: Practice classifying questions, understand sampling methods, memorize variability measures, interpret data meaningfully.
Formulas to know by heart:

• Range Formula: Range = Maximum Value - Minimum Value

• Statistical Question Principle: Anticipates variability in responses

• Population vs Sample: Sample is subset of population

• Variability Assessment: Larger, more diverse groups show greater variability

Exercise with Visualization: Data Distribution
Exercise 6: Comparing Data Sets
Consider the following statistical questions and their expected data distributions:
Question A: "How many pets do families in the neighborhood have?"
Question B: "How many siblings do students in class have?"
Question C: "How tall are the trees in the park?"

Analysis: The visualization shows how different statistical questions produce different data distributions.

  • Question A: Likely has low variability (most families have 0-3 pets)
  • Question B: Moderate variability (students have 0-5+ siblings)
  • Question C: High variability (trees vary greatly in height)

Questions & Answers

Question: How can I tell if a question is statistical or not? Sometimes I get confused.

Answer: Here's a foolproof method to identify statistical questions:

  • Ask yourself: "Will different people give different answers?" If yes, it's statistical.
  • Look for groups: Questions about groups (students, adults, families) are usually statistical.
  • Check for variability: Does the question expect answers to vary? If so, it's statistical.
  • One answer test: If there's only one correct answer, it's non-statistical.

Examples: "How old are the students?" → Statistical (varies by student)

"How old is the teacher?" → Non-statistical (one answer)

Remember: Statistical questions are about collecting and analyzing data from multiple sources!

Question: What's the difference between population and sample? Can you give me a real-life example?

Answer: Think of it this way:

  • Population: The entire group you want to learn about
  • Sample: The smaller group you actually study to learn about the population

Real-life example: A pizza restaurant wants to know what toppings customers prefer.

  • Population: ALL customers who visit the restaurant
  • Sample: 100 randomly selected customers who fill out a survey

The restaurant studies the sample to make conclusions about the population. It's impractical to ask every single customer, so they use a representative sample.

The sample should reflect the diversity of the population to ensure accurate results.

Question: Why does variability matter in statistics? Isn't it better to have consistent data?

Answer: Variability is actually essential in statistics! Here's why:

  • Real-world reflection: Most natural phenomena have inherent variability
  • Information content: High variability can reveal important patterns and differences
  • Statistical significance: Helps determine if observed differences are meaningful
  • Decision making: Understanding variability leads to better decisions

For example, if you're comparing test scores between two classes, you need to consider variability. If Class A has scores of 70, 80, 90 and Class B has scores of 80, 80, 80, both have the same average (80), but Class A has more variability.

This tells you that Class A has more diverse performance levels, while Class B is more consistent. Both pieces of information are valuable!

Question: What are some common ways that statistical questions can be biased or misleading?

Answer: Here are common sources of bias in statistical questions:

  • Leading questions: Phrasing that influences responses ("Don't you think this is the best option?")
  • Convenience sampling: Only surveying easily accessible people
  • Small sample size: Not having enough participants to represent the population
  • Non-random selection: Systematically excluding certain groups
  • Wording bias: Using emotionally charged language

For example, asking "How much do you love our excellent school?" is leading, while "How would you rate your satisfaction with our school?" is neutral.

Always aim for random, representative samples and neutral question wording to minimize bias and ensure accurate results.

Question: How can I check if my understanding of a statistical question is correct?

Answer: Here are effective verification strategies:

  1. Variability Test: Imagine asking the question to different people - will their answers vary?
  2. Population Check: Does the question refer to a group rather than an individual?
  3. Answer Plausibility: Is it reasonable to expect multiple different answers?
  4. Context Clue: Look for words like "average," "typical," "range," or "distribution"

Also consider: Would collecting data from this question require a survey or measurement of multiple subjects?

Practice with many examples to build your intuition. Over time, you'll develop a strong sense of what constitutes a statistical question.

Remember: The key characteristic is anticipation of variability in the data!