t-Distribution vs Normal
Examine the heavy-tailed Student's t-distribution. Increase degrees of freedom (df) to watch it perfectly converge into the standard normal z-distribution.
WHEN WE DON'T KNOW SIGMA
In the real world, we rarely know the true population standard deviation (). Instead, we must use the sample standard deviation (). This adds uncertainty, which we account for using the **t-distribution**.
SHAPE AND TAILS
The t-distribution is symmetric and bell-shaped like the normal distribution, but it has "heavier tails." This means there is more area far from the center, reflecting the extra variability of using .
AP EXAM CONNECTION
Unit: Unit 7: Inference for Means (Topic 7.2)
Learning Objective: UNC-4.D
COMMON MISCONCEPTIONS
- Using z-procedures when the population standard deviation is unknown.
KEY TAKEAWAYS
- Used for means.
- Heavier tails than normal.
- df = n - 1.
PRACTICE QUESTIONS
Q1 (CONCEPTUAL): What happens to the t-distribution as the degrees of freedom increase?
Show Answer & Explanation
Answer: It approaches the normal distribution.
Explanation: As sample size increases, our estimate of the standard deviation becomes more precise, and the t-distribution converges to the z-distribution.
