Domain: Problem-Solving and Data Analysis | Skill: Inference from sample statistics and margin of error | Difficulty: Hard
Master Statistical Inference: Tackling Hard SAT Questions on Sample Statistics and Margin of Error
When it comes to the most challenging questions on the SAT Math section, statistical inference problems involving sample statistics and margin of error often separate top scorers from the rest. These questions test your ability to understand how sample data relates to population parameters, interpret confidence intervals, and make sophisticated comparisons between different statistical scenarios. At the hard difficulty level, you’ll encounter complex situations requiring deep conceptual understanding rather than just formula application.
Question Types You’ll Encounter
| Typical Format | What It Tests | Quick Strategy |
|---|---|---|
| Comparing confidence intervals from different samples | Understanding how sample size affects margin of error | Remember: larger samples → smaller margins of error |
| Finding required sample size for a specific margin of error | Inverse relationship between sample size and precision | Use the formula: margin of error ∝ 1/√n |
| Interpreting overlapping confidence intervals | Statistical significance and practical conclusions | Overlapping intervals don’t always mean no difference |
| Combined samples and pooled estimates | Weighted averages and variance calculations | Weight by sample size when combining results |
Real SAT-Style Example
Question: Two independent surveys were conducted to estimate the proportion of residents in a town who support a new park proposal. The first survey sampled 400 residents, and 240 of them expressed support. The second survey sampled 600 residents, and 330 expressed support. The results of each survey are used to construct a 95% confidence interval for the true proportion of residents who support the proposal. Let \(w_1\) be the width of the confidence interval based on the first survey, and \(w_2\) be the width of the confidence interval based on the second survey. Which of the following best describes the relationship between \(w_1\) and \(w_2\)?
A) \(w_1 < w_2\)
B) \(w_1 = w_2\)
C) \(w_1 > w_2\) ✅
D) The relationship between \(w_1\) and \(w_2\) cannot be determined from the information given.
Solution:
First, let’s find the sample proportions:
Survey 1: \(\hat{p}_1 = \frac{240}{400} = 0.6\)
Survey 2: \(\hat{p}_2 = \frac{330}{600} = 0.55\)
The width of a confidence interval for a proportion is:
\[w = 2 \times z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]
For Survey 1:
\[w_1 = 2 \times 1.96 \times \sqrt{\frac{0.6 \times 0.4}{400}} = 3.92 \times \sqrt{\frac{0.24}{400}} = 3.92 \times 0.0245 \approx 0.096\]
For Survey 2:
\[w_2 = 2 \times 1.96 \times \sqrt{\frac{0.55 \times 0.45}{600}} = 3.92 \times \sqrt{\frac{0.2475}{600}} = 3.92 \times 0.0203 \approx 0.080\]
Since \(0.096 > 0.080\), we have \(w_1 > w_2\).
The answer is C.
Step-by-Step Strategy for Hard Statistical Inference Questions
- Identify the statistical concepts involved – Determine whether you’re dealing with proportions, means, confidence intervals, or hypothesis testing.
- Extract and organize the given information – List sample sizes, statistics, confidence levels, and what you need to find.
- Apply the appropriate formula or principle – Remember key relationships like margin of error being inversely proportional to square root of sample size.
- Perform calculations strategically – Often you can compare relative sizes without computing exact values.
- Verify your answer makes intuitive sense – Larger samples should give more precise estimates, higher confidence levels mean wider intervals, etc.
Applying the Strategy to Our Example
Step 1 Applied: Identify the Concepts
We’re comparing confidence interval widths for two proportion estimates from different sample sizes. The key concept is how sample size affects the margin of error.
Step 2 Applied: Extract Information
Survey 1: n₁ = 400, x₁ = 240 supporters
Survey 2: n₂ = 600, x₂ = 330 supporters
Both use 95% confidence intervals
Need to compare w₁ and w₂
Step 3 Applied: Apply the Formula
Width formula: \(w = 2 \times z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)
Notice that the z-value is the same for both (1.96 for 95% CI)
The key difference will be in the standard error term
Step 4 Applied: Strategic Calculation
Instead of calculating exact widths, we can compare the standard error terms:
For Survey 1: \(\sqrt{\frac{0.6 \times 0.4}{400}} = \sqrt{\frac{0.24}{400}}\)
For Survey 2: \(\sqrt{\frac{0.55 \times 0.45}{600}} = \sqrt{\frac{0.2475}{600}}\)
Since 0.24/400 = 0.0006 and 0.2475/600 = 0.0004125, Survey 1 has the larger standard error.
Step 5 Applied: Verify Intuition
Survey 2 has a larger sample size (600 vs 400), so it should have a smaller margin of error and thus a narrower confidence interval. This confirms w₁ > w₂.
Key Takeaways
- Larger sample sizes always lead to smaller margins of error and narrower confidence intervals
- The margin of error is proportional to \(\frac{1}{\sqrt{n}}\), not \(\frac{1}{n}\)
- When comparing intervals, you often don’t need exact calculations—relative comparisons work
- Understanding the conceptual relationships is more important than memorizing formulas for hard questions
- Practice interpreting what statistical results mean in context, not just calculating them
