8. Associations and Causation

Math 115 - Examples for Topic 8

Author

Yurk

These worked examples cover contingency tables, conditional proportions, and scope of inference (generalization and causation).

Example 1: Conditional Proportions

A marketing research firm surveyed 600 adults about their preferred streaming service. The results are summarized in the contingency table below, broken down by age group.

Age Group Netflix Disney+ HBO Max Total
Under 30 145 85 50 280
30 and over 120 70 130 320
Total 265 155 180 600

Part A

Calculate the proportion of adults under 30 who prefer Netflix.

We want the proportion of adults under 30 who prefer Netflix.

This is a row-conditioned proportion: we’re looking within the “Under 30” row.

\[\text{Proportion} = \frac{\text{Under 30 who prefer Netflix}}{\text{Total Under 30}} = \frac{145}{280} = 0.518\]

Answer: About 51.8% of adults under 30 prefer Netflix.

Part B

Calculate the proportion of adults 30 and over who prefer HBO Max.

We want the proportion of adults 30 and over who prefer HBO Max.

This is also a row-conditioned proportion: we’re looking within the “30 and over” row.

\[\text{Proportion} = \frac{\text{30 and over who prefer HBO Max}}{\text{Total 30 and over}} = \frac{130}{320} = 0.406\]

Answer: About 40.6% of adults 30 and over prefer HBO Max.

Part C

Calculate the proportion of Netflix users who are under 30.

We want the proportion of Netflix users who are under 30.

This is a column-conditioned proportion: we’re looking within the “Netflix” column.

\[\text{Proportion} = \frac{\text{Netflix users Under 30}}{\text{Total Netflix users}} = \frac{145}{265} = 0.547\]

Answer: About 54.7% of Netflix users are under 30.

Part D

A colleague provides you with the following table of proportions. Determine whether these proportions are conditioned on rows (age group) or columns (streaming service). Explain your reasoning.

Age Group Netflix Disney+ HBO Max
Under 30 0.518 0.304 0.179
30 and over 0.375 0.219 0.406

To determine whether the proportions are conditioned on rows or columns, we can check which way the proportions sum to 1.

Check row sums:

  • Under 30: \(0.518 + 0.304 + 0.179 = 1.001 \approx 1\) ✓
  • 30 and over: \(0.375 + 0.219 + 0.406 = 1.000\) ✓

Check column sums:

  • Netflix: \(0.518 + 0.375 = 0.893 \neq 1\)
  • Disney+: \(0.304 + 0.219 = 0.523 \neq 1\)
  • HBO Max: \(0.179 + 0.406 = 0.585 \neq 1\)

Answer: The proportions are conditioned on rows (age group). We can tell because the proportions in each row sum to 1 (or approximately 1, accounting for rounding), while the column sums do not equal 1.

This means each proportion represents the distribution of streaming service preferences within each age group. For example, 0.518 means that 51.8% of people under 30 prefer Netflix.

Example 2: Scope of Inference

For each of the following scenarios involving two proportions, determine:

  • Can the results be generalized to a broader population?
  • Can a cause-and-effect relationship be established?

Explain your reasoning for each answer.

Scenario A

A tech company wants to test whether a new website design increases the proportion of visitors who sign up for their newsletter. They take all 2,000 visitors to their website during one week and randomly assign half to see the old design and half to see the new design. They find that 12% of visitors seeing the new design signed up, compared to 8% with the old design.

Setting: A/B test with all website visitors randomly assigned to design groups.

Can the results be generalized?

No. The 2,000 visitors during that one week were not randomly selected from any broader population—they are simply whoever happened to visit the website that week. These visitors may not be representative of all potential website visitors (different times of year, different marketing campaigns, etc.).

Can cause-and-effect be established?

Yes. This is a randomized experiment. Visitors were randomly assigned to see either the old or new design. Random assignment ensures that the two groups are similar on average with respect to all other characteristics (device type, time of day visiting, prior familiarity with the company, etc.). Therefore, the difference in signup rates can be attributed to the website design itself.

Summary: Causal, but not generalizable.

Scenario B

Researchers want to study whether there is a difference in the proportion of college students who experience anxiety between those who use social media more than 3 hours per day versus those who use it less. They randomly select 500 students from a large university’s enrollment records and survey them about their social media use and anxiety levels. They find that 45% of heavy social media users report anxiety, compared to 28% of lighter users.

Setting: Random sample of students, observed (not assigned) for social media use.

Can the results be generalized?

Yes. The 500 students were randomly selected from the university’s enrollment records. This random sampling means the results can be generalized to all students at that university.

Can cause-and-effect be established?

No. This is an observational study. Students were not randomly assigned to be heavy or light social media users—they self-selected into these groups based on their own behaviors and preferences. There could be confounding variables that explain the association. For example, students who are already prone to anxiety might be more likely to use social media heavily as a coping mechanism, or students with heavier course loads might have both more stress and more need for social media breaks.

Summary: Generalizable (to students at this university), but association only—not causal.

Scenario C

A hospital wants to determine whether a new medication reduces the proportion of patients who experience post-surgical complications. They randomly select 200 patients from all patients scheduled for a specific surgery nationwide, then randomly assign 100 to receive the new medication and 100 to receive standard care. They find that 5% of patients on the new medication had complications, compared to 15% with standard care.

Setting: Random sample of patients AND random assignment to treatment groups.

Can the results be generalized?

Yes. Patients were randomly selected from all patients scheduled for this surgery nationwide. This random sampling means the results can be generalized to the broader population of patients undergoing this surgery.

Can cause-and-effect be established?

Yes. This is a randomized experiment. Patients were randomly assigned to receive the new medication or standard care. Random assignment ensures the treatment groups are similar on average with respect to potential confounders (age, overall health, surgical complexity, etc.). The difference in complication rates can be attributed to the medication.

Summary: Both causal AND generalizable. This is the gold standard of study design.

Scenario D

A local coffee shop owner notices that customers who order pastries seem more likely to return the next week than those who don’t. To investigate, she reviews purchase records for 150 customers who visited during the past month. She finds that 60% of pastry buyers returned within a week, compared to 35% of non-pastry buyers.

Setting: Convenience sample of existing customers, observed (not assigned) for pastry purchasing.

Can the results be generalized?

No. The 150 customers were not randomly selected from any defined population—they are simply whoever happened to visit that specific coffee shop during the past month. These customers may not be representative of coffee shop customers in general, or even of all potential customers of this shop.

Can cause-and-effect be established?

No. This is an observational study. Customers were not randomly assigned to buy or not buy pastries—they chose for themselves. There could be confounding variables. For example, customers who are regulars might be more likely to both buy pastries (because they know the menu well) and return soon (because they’re already loyal customers). Or customers celebrating special occasions might buy pastries as a treat and also be more likely to associate the shop with positive memories.

Summary: Neither generalizable nor causal—association only in this specific sample.

Summary Table

Scenario Random Sampling? Random Assignment? Generalizable? Causal?
A (Website A/B test) No Yes No Yes
B (Social media survey) Yes No Yes No
C (Medication trial) Yes Yes Yes Yes
D (Coffee shop records) No No No No
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