Module 1 Practice Problems

Solution:

(a) Descriptive Statistics

Explanation: The teacher is simply summarizing data she already has (her class's grades). She's not making predictions or generalizations beyond her specific class.

(b) Inferential Statistics

Explanation: The news outlet is using data from a sample (500 voters) to make a prediction about a larger population (all voters). They're inferring from the sample to the population.

(c) Descriptive Statistics

Explanation: The hospital is describing what happened—summarizing the births at their specific location. No predictions or generalizations are being made.

(d) Inferential Statistics

Explanation: The company tested a sample (2,000 patients) and is drawing conclusions about the general population. They're using sample data to infer what will happen for everyone.

Key Concept: Descriptive statistics summarize what you have. Inferential statistics predict or generalize beyond your data.

Sample Answers:

Example 1: Weather Forecast

Statistic: "70% chance of rain tomorrow"

What it measures: The probability (likelihood) that it will rain based on historical weather patterns and current conditions

Decision impact: I decided to bring an umbrella to campus because 70% is a high probability

Example 2: Product Reviews

Statistic: "4.5 stars based on 1,247 reviews" for a laptop I was considering

What it measures: Average customer satisfaction based on ratings from 1,247 people who purchased the laptop

Decision impact: The high rating (4.5/5) and large number of reviews (1,247) gave me confidence to purchase that laptop over others with fewer reviews

Example 3: Social Media Engagement

Statistic: "Your post reached 523 people" on Instagram

What it measures: The number of unique users who saw my post in their feed

Decision impact: Seeing this engagement stat made me post more content at that time of day when my posts get more reach

Note: Your examples will be different! The point is to recognize how often you encounter statistics and how they influence decisions.

Solution:

(a) No, we cannot conclude that buying books causes higher income.

Reasoning: This is an observational study (not an experiment), so we can't establish causation. Just because two things are correlated doesn't mean one causes the other. It's unlikely that simply purchasing books would increase your income!

(b) Possibly, but we still can't prove causation from this study alone.

Reasoning: While it's more plausible that higher income → buying more books (you have more disposable income), we still only have correlation, not causation. We'd need an experiment to prove it.

(c) Alternative explanations (confounding variables):

• Education level: People with more education tend to have higher incomes AND tend to own more books. Education could be causing both!
• Reading interest/culture: People who value learning may both pursue higher-paying careers and collect books
• Age: Older people have had more time to accumulate both wealth and book collections
• Location: People in cities may have higher incomes and more access to bookstores

Key Lesson: Correlation ≠ Causation. There's almost always a "third variable" that could explain the relationship!

Solution:

(a) Quantitative - Discrete

Reasoning: Number of siblings is numerical (quantitative) and can only be whole numbers (0, 1, 2, 3...). You can't have 2.5 siblings!

(b) Qualitative (Categorical)

Reasoning: Brand names are categories, not numbers. Can't do math with "Apple + Samsung"!

(c) Quantitative - Continuous

Reasoning: Time is numerical (quantitative) and can take any value—2.5 hours, 2.53 hours, etc. It's measured, not counted.

(d) Qualitative (Categorical - Ordinal)

Reasoning: T-shirt sizes are categories. Even though there's an order (S < M < L < XL), they're not numerical values.

(e) Quantitative - Continuous

Reasoning: Heart rate is numerical and can be measured precisely—72 bpm, 72.3 bpm, etc.

(f) Quantitative - Discrete

Reasoning: Number of courses is numerical and can only be whole numbers (1, 2, 3, 4 courses, not 3.7 courses).

(g) Quantitative - Continuous

Reasoning: Temperature is numerical and can take any value (72.5°F, 72.53°F, etc.).

(h) Qualitative (Categorical - Nominal)

Reasoning: Marital status categories have no inherent numerical meaning or order.

Quick Check: If you can COUNT it (only whole numbers) → Discrete. If you MEASURE it (any value) → Continuous.

Solution:

(a) Ordinal

Justification: The ratings have a meaningful order (terrible < poor < good < excellent), but the "distance" between ratings isn't necessarily equal. The jump from terrible to poor might feel different than poor to good. You can rank them but can't do meaningful math (can't say "excellent is twice as good as poor").

(b) Ratio

Justification: Distance has equal intervals (1 mile = 1 mile everywhere) AND a true zero (0 miles = no distance driven). You can meaningfully say "10 miles is twice as far as 5 miles." This is the highest level of measurement—all math operations are valid!

(c) Interval

Justification: Years have equal intervals (2024-2023 = same as 2025-2024), but there's no true "zero year" that means "absence of time." Year 0 is arbitrary (we use it for the calendar, but time existed before!). You can't say "2024 is twice as much as 1012."

(d) Nominal

Justification: Student ID numbers are just labels/names. Even though they're written as numbers, they have no mathematical meaning. ID #5000 isn't "bigger" or "better" than ID #2500—they're just identifiers. No order, no intervals, no zero point.

Summary Hierarchy: Nominal < Ordinal < Interval < Ratio (each level has all properties of the ones below it plus more)

Solution:

All three are qualitative (categorical) even though they're written as numbers!

(a) Zip Codes

Why qualitative: Zip codes are geographic identifiers—labels for locations, not quantities. You can't meaningfully average zip codes or say 90405 is "greater than" 90402. Even though they're digits, they function as categories (like names).

(b) Jersey Numbers

Why qualitative: Jersey numbers identify players—they're labels, not measurements. A player wearing #23 isn't "better than" or "more than" a player wearing #12. You can't average jersey numbers meaningfully. They're categorical identifiers.

(c) Social Security Numbers

Why qualitative: SSNs are unique identifiers for people. They have no numerical meaning. You wouldn't say someone with SSN 555-12-3456 is "greater than" someone with SSN 123-45-6789. They're nominal categorical data—just using numbers as labels.

The Test: Ask yourself: "Can I do meaningful math with these numbers?" If adding, averaging, or comparing them doesn't make sense, they're probably qualitative labels, not quantitative measurements!

Solution:

(a) Experiment

Reasoning: Researchers are actively manipulating a variable (medication vs. placebo) and using random assignment. This is the hallmark of an experiment. Only experiments can establish causation!

(b) Observational Study

Reasoning: The psychologist is watching and recording what happens naturally without interfering or manipulating anything. Classic observational study.

(c) Survey

Reasoning: The company is asking people questions to collect data about their opinions/experiences. This is a survey (questionnaire method).

(d) Observational Study

Reasoning: The scientists are observing and recording data that occurs naturally (temperature and sales) without manipulating anything. They're looking for an association, not testing causation.

Key Distinction: If researchers actively manipulate/control something → Experiment. If they just observe/ask → Observational/Survey.

Solution:

(a) Simple Random Sample

Reasoning: Every student has an equal chance of being selected from the entire population. This is the gold standard for unbiased sampling!

(b) Cluster Sample

Reasoning: The population was divided into natural groups (dorms), entire clusters were randomly selected, and everyone in those clusters was surveyed. Classic cluster sampling!

(c) Convenience Sample

Reasoning: This is BIASED sampling! They're just surveying whoever is easy to reach (first people in the dining hall). This group might not represent all students—what about students who don't eat at the dining hall, or who come at different times?

(d) Stratified Random Sample

Reasoning: The population was divided into strata (class years), and random samples were taken from each stratum to ensure all groups are represented equally.

Which is best? (a), (b), and (d) are all valid methods. Method (c) is problematic because it's likely biased—avoid convenience sampling in research!

Solution:

(a) Three Sources of Bias:

1. Voluntary Response Bias: People self-select whether to respond
2. Non-Response Bias: Only 200 responded (out of how many members?)
3. Selection Bias: Only members who visit the gym's website see the survey

(b) How Each Bias Affects Results:

1. Voluntary Response Bias: People with strong opinions (very happy OR very unhappy) are more likely to respond. However, in this case, people who are SATISFIED are probably more willing to take time to fill out their gym's survey, while dissatisfied members might have already quit or ignore the survey. This could inflate the satisfaction rate.

2. Non-Response Bias: The 85% satisfaction only reflects the 200 people who responded. What about the thousands who didn't respond? They might be less satisfied, too busy, or disengaged. We're missing a huge portion of the membership's opinions.

3. Selection Bias: Only people who regularly check the gym's website will see the survey. Members who don't use the website (perhaps less engaged members?) never get the chance to respond. This excludes certain groups entirely.

(c) Better Data Collection Method:

Recommended approach: Stratified Random Sampling with follow-up

• Get a complete list of all current gym members
• Randomly select 300-500 members (ensuring different age groups and membership types are represented)
• Contact them via email AND text with survey link
• Follow up with non-responders with phone calls or in-person requests
• Aim for at least 60-70% response rate
• Consider offering a small incentive (raffle entry) to increase response rate

Why this is better: Random selection reduces bias, stratification ensures all groups are represented, and follow-up reduces non-response bias. You get a more accurate picture of true member satisfaction!

Solution:

(a) Identifying the Methods:

Option A: Observational Study (Survey) - Just asking and observing, no manipulation

Option B: Experiment - Actively assigning treatments (study techniques) to groups

(b) Which Can Establish Causation?

Option B (Experiment) can establish causation.

Why: Because of random assignment! When you randomly assign students to groups, those groups should be similar in every way except the study technique. They should have similar:

• Prior knowledge and ability
• Motivation levels
• Time available for studying
• Test anxiety levels
• Any other factors that could affect scores

If the new technique group scores higher, it's reasonable to conclude the technique caused the improvement, since the groups were otherwise identical!

Option A (Observational) cannot establish causation because students chose their own study techniques—they weren't randomly assigned. Differences in scores could be due to the technique OR due to differences in the students themselves.

(c) Confounding Variables in Option A:

• Student motivation: More motivated students might both seek out new study techniques AND study more hours, leading to higher scores
• Prior academic performance: Better students might be more likely to experiment with new techniques
• Time availability: Students with more free time might use more elaborate study techniques
• Learning style: Different techniques might work better for different learners
• Course difficulty: Students in different sections or majors might face different exam difficulty levels

Why Option B Avoids These: Random assignment distributes all these confounding variables evenly across both groups. Some motivated students will be in each group, some struggling students in each group, etc. The technique is the ONLY systematic difference!

Bottom Line: If you want to prove X causes Y, you need a well-designed experiment with random assignment. Observational studies can show correlation, but never causation!

Solution:

(a) Line Graph

Why: Population is changing over TIME (1950-2020). Line graphs are perfect for showing trends over time. The line connecting the points shows the continuous growth pattern.

(b) Bar Chart

Why: We're comparing quantities across different CATEGORIES (the four majors). Bar charts make it easy to compare the heights of bars to see which major has more students.

(c) Histogram

Why: Height is continuous numerical data, and we want to see the DISTRIBUTION (shape) of the data. A histogram groups heights into ranges (bins) like 5'0"-5'2", 5'2"-5'4", etc., and shows how many students fall in each range. This reveals whether heights are normally distributed, skewed, etc.

(d) Scatterplot

Why: We want to see the RELATIONSHIP between two quantitative variables (hours slept and test score). Each point represents one student with their sleep hours on x-axis and test score on y-axis. The pattern of points reveals whether there's a correlation.

(e) Pie Chart

Why: We're showing PARTS OF A WHOLE where the percentages add to 100%. A pie chart visually shows how the total budget is divided among categories. (Note: A bar chart would also work here and might be easier to read!)

Solution:

(a) Total number of students: 50 students

Calculation: Add all the frequencies: 2 + 5 + 18 + 20 + 5 = 50

(b) Range with most students: 80-90 (20 students)

Reasoning: This bin has the tallest bar with 20 students.

(c) Distribution shape: Skewed left (negatively skewed)

Reasoning: The bulk of the data is on the RIGHT (higher scores), with a tail extending to the LEFT (lower scores). Most students did well (70-90 range), but a few students scored lower, creating the left tail. Think of it as: the tail points to the skew direction!

(d) Percentage who scored 80% or higher: 50%

Calculation:

• Students with 80% or higher: 20 (from 80-90) + 5 (from 90-100) = 25 students
• Percentage: (25 / 50) × 100% = 50%

Half the class scored 80% or higher!

Histogram Reading Tips:

• Add up all bar heights to get total count
• The tallest bar is the mode (most common range)
• Where the tail points = direction of skew
• Sum specific bins to answer "how many" or "what percent" questions

Solution:

(a) Actual percentage increase:

Calculation:

• Increase amount: $102,000 - $98,000 = $4,000
• Percentage: ($4,000 / $98,000) × 100% ≈ 4.08%

That's about a 4% increase—modest, not "skyrocketing"!

(b) Why it's misleading:

When the y-axis starts at $95,000 instead of zero:

• The 2023 bar only goes from $95k to $98k (3 units tall)
• The 2024 bar goes from $95k to $102k (7 units tall)
• Visually, it looks like sales more than DOUBLED (7/3 = 2.33x taller)
• In reality, sales increased by only 4%!

The truncated axis exaggerates the visual difference, creating a false impression of dramatic growth.

(c) How it would look with y-axis at zero:

If the y-axis started at $0:

• 2023 bar: 98 units tall
• 2024 bar: 102 units tall
• The bars would look almost the same height (because they ARE almost the same!)
• The 4% increase would be visible but appropriately small

This would accurately represent the modest growth.

(d) When is NOT starting at zero acceptable?

LINE GRAPHS (sometimes acceptable):

• When tracking small changes in large values over time (e.g., daily stock prices from $150-$155)
• When the focus is on TREND rather than total magnitude
• When the y-axis is clearly labeled so readers understand the scale
• Example: A line graph of body temperature over a day (98°F-101°F) doesn't need to start at 0°F

BAR CHARTS (rarely acceptable):

• Bar charts should ALMOST ALWAYS start at zero because they show magnitude/quantity
• The length of the bar represents the value, so truncating distorts the comparison

Best practice: When in doubt, start at zero. If you must truncate, include a "break" symbol in the axis () to show the scale is not starting at zero, and make this VERY clear to readers!

Solution:

Feature	Bar Chart	Histogram
Data Type	Categorical (qualitative) data	Continuous quantitative data
Do bars touch?	No - bars have gaps between them	Yes - bars touch each other
Why?	Gaps show categories are distinct and separate	Touching bars show data is continuous (no gaps in values)
X-axis shows	Category names (Ice cream flavors, Majors, etc.)	Numerical ranges/bins (0-10, 10-20, 20-30, etc.)
Y-axis shows	Frequency or count for each category	Frequency (how many values fall in each bin)
Example Use	Comparing number of students in different majors (Biology: 150, Chemistry: 120, Physics: 90)	Showing distribution of student ages (18-19: 500 students, 19-20: 600 students, 20-21: 450 students)

Key Memory Trick:

Bar chart = Bars have Breaks (gaps) → for Categories

Histogram = bars Hug each other (touch) → for Continuous data

Visual Difference Example:

Bar Chart (Favorite Colors):

[Red]___[Blue]___[Green]___[Yellow]

← Notice gaps between bars for distinct categories

Histogram (Test Scores):

[50-60][60-70][70-80][80-90][90-100]

← Notice bars touching because scores are continuous

Solution:

(a) Graph type: Scatterplot

Why: You want to see the RELATIONSHIP between two quantitative variables (study hours and GPA). A scatterplot displays each student as a point, with one variable on each axis, revealing any correlation patterns.

(b) Axis assignment:

• X-axis: Study hours per week (independent/explanatory variable)
• Y-axis: GPA (dependent/response variable)

Why this order: We're investigating whether study hours might affect GPA, so hours go on x-axis and GPA on y-axis. Convention is: if one variable might influence the other, the influencing variable goes on x.

(c) 5 Best Practices:

1. Clear, descriptive title: "Relationship Between Study Hours and GPA for 50 College Students"
2. Label both axes with units: X-axis: "Study Hours per Week", Y-axis: "GPA (0.0-4.0 scale)"
3. Use appropriate scales: Start axes at reasonable values (GPA from 0.0-4.0, study hours from 0)
4. Make points visible: Use clear markers/dots that don't overlap too much; if overlap is an issue, use semi-transparent points
5. Include sample size and source: Note "n=50 students" and "Data collected: Fall 2024"

Bonus practices:

• Keep it simple—no unnecessary decorations or 3D effects
• Use color purposefully (or keep it monochrome)
• Ensure axes have reasonable increments (not too many tick marks, not too few)

(d) Interpretation:

What upward trend DOES tell you:

• There is a positive correlation between study hours and GPA
• Students who study more hours tend to have higher GPAs
• As study hours increase, GPA tends to increase too
• There's an association between the two variables

What upward trend does NOT tell you:

• It does NOT prove causation! You can't conclude that studying more causes higher GPA
• Why not? This is observational data, not an experiment. Many confounding variables could explain the relationship:
  • More motivated students might both study more AND have higher ability
  • Students taking easier courses might have more time to study and higher grades
  • Better study techniques (not just hours) might explain the difference
  • Prior knowledge from high school might affect both study efficiency and grades
• The correlation doesn't tell you the exact GPA for any given study hours (it's a trend, not a perfect prediction)
• It doesn't tell you if the relationship is linear or curved

Bottom line: A scatterplot can reveal correlation, but remember: Correlation ≠ Causation! To prove studying more causes higher GPA, you'd need a controlled experiment.