Foundations of Statistical Inference

Why We Need Statistical Inference

• Suppose women give Dems an average rating of 54, men give 49.
• Is the 5-point gap real or just sampling noise?
• Inference tells us: how confident can we be that our result reflects the population?

Key Concepts

• Population: All cases (e.g., all U.S. voters)
• Parameter: True population value (unknown)
• Sample: Subset we actually study
• Statistic: Estimate of the parameter based on the sample

Why Random Sampling Matters

Only random samples give every member of the population an equal chance of selection.

Random Sampling Error

• Even with good samples, estimates vary.
• This variation is random sampling error, not a mistake.

The Standard Error

• Measures how much a statistic is expected to vary due to sampling error.
• Decreases as sample size increases.

Central Limit Theorem

• Sampling distributions of means form a bell curve.
• Holds regardless of population shape, if n is large enough.

Z Scores and the Normal Curve

• A Z score = how many SEs away a value is from the mean
• Helps us use the empirical rule:
  • 68% within 1 SE
  • 95% within 2 SEs

Confidence Intervals

• 95% CI = sample statistic ± 1.96 × SE
• Shows range where we expect population value to fall

Sample Size and Diminishing Returns

• More data = less error, but benefits shrink with size

The t-Distribution for Small Samples

• For small samples, use the t-distribution instead of normal
• Thicker tails, wider CIs

Wrap-Up

• Random sampling lets us infer from sample to population
• Standard error and confidence intervals express uncertainty
• Larger samples and less variation = more precise estimates
• Use t-distribution when sample sizes are small