Confidence Intervals for Means

Why Not z? The t-Distribution

When we estimate a population mean, we usually don't know $\sigma$, so we estimate it with $s$ (sample standard deviation). This introduces extra uncertainty, so we use the t-distribution instead of the Normal distribution.

One-Sample t-Interval for $\mu$

$\bar{x} \pm t^* \frac{s}{\sqrt{n}}$

where:

• $\bar{x}$ = sample mean
• $t^*$ = critical value from the $t$-distribution with $df = n - 1$
• $\frac{s}{\sqrt{n}}$ = standard error of $\bar{x}$

The t-Distribution

Properties:

• Symmetric and bell-shaped (like Normal)
• More spread out than the Normal (heavier tails)
• Characterized by degrees of freedom ($df$)
• As $df \to \infty$, the $t$-distribution approaches the Normal

Conditions

1. Random: Data from a random sample or randomized experiment
2. 10% Condition: $n < 0.10N$
3. Normal/Large Sample: Population is Normal, OR $n \geq 30$ (CLT)
   • For $n < 30$: Check for strong skewness or outliers (use graphs)
   • If the sample has extreme outliers, the t-interval is not reliable

Interpretation

"We are [C]% confident that the true mean [context] is between [lower] and [upper]."

Paired t-Interval

For matched pairs data (before/after, twin studies):

1. Calculate the differences $d_i = x_{1i} - x_{2i}$
2. Apply the one-sample t-interval to the differences

$\bar{d} \pm t^* \frac{s_d}{\sqrt{n}}$

t-Table Values (Selected)

| df | 90% ($t^*$) | 95% ($t^*$) | 99% ($t^*$) | |----|------------|------------|------------| | 5 | 2.015 | 2.571 | 4.032 | | 10 | 1.812 | 2.228 | 3.169 | | 20 | 1.725 | 2.086 | 2.845 | | 30 | 1.697 | 2.042 | 2.750 | | $\infty$ | 1.645 | 1.960 | 2.576 |

Choosing Between z and t

| Situation | Use | |-----------|-----| | $\sigma$ known (rare) | z-interval | | $\sigma$ unknown | t-interval | | Proportion | z-interval | | Mean | t-interval |

AP Tip: On the AP exam, always use the t-distribution for means unless explicitly told $\sigma$ is known. The z-interval for means is almost never used in practice.