📏 Confidence Intervals for Means

When to Use t-Interval vs z-Interval

• Use t-interval: When σ (population SD) is unknown (almost always in practice)
• Use z-interval: Only when σ is known (rare in real applications)

One-Sample t-Interval Formula

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

where:

• $\bar{x}$ = sample mean
• $t^*$ = critical t-value (depends on confidence level and degrees of freedom)
• $s$ = sample standard deviation
• $n$ = sample size
• $df = n - 1$

Conditions for t-Interval

1. Random sample: Data collected randomly
2. Independence: Sampling without replacement; use 10% rule (n ≤ 0.10N)
3. Normality: Either population is normal OR n ≥ 30 (CLT)

Finding $t^*$ Values

• df = n − 1
• Look up $t^*$ in t-table using df and confidence level

Common values (df = large, approximately normal):

• 90% CI: $t^* \approx 1.645$
• 95% CI: $t^* \approx 1.96$
• 99% CI: $t^* \approx 2.576$

For small samples:

• df = 9, 95% CI: $t^* = 2.262$ (larger than z)
• df = 4, 95% CI: $t^* = 2.776$ (even larger)

Smaller df → larger $t^*$ → wider CI.

One-Sample Example

A random sample of 16 students has mean test score $\bar{x} = 78$ with sample SD s = 8. Find a 95% CI for the population mean.

Check conditions:

• Random sample ✓
• n = 16 < 30, but assume population approximately normal ✓
• Independence ✓

Calculate:

• $df = 16 - 1 = 15$
• From t-table: $t^* = 2.131$ (95% CI, df = 15)
• $SE = \frac{8}{\sqrt{16}} = \frac{8}{4} = 2$
• $ME = 2.131 \times 2 = 4.262$
• CI: $78 \pm 4.262 = (73.738, 82.262)$

Interpretation: We are 95% confident the mean score is between 73.7 and 82.3.

Two-Sample t-Interval

Comparing two population means $\mu_1$ and $\mu_2$:

$(\bar{x}_1 - \bar{x}_2) \pm t^* \cdot SE(\bar{x}_1 - \bar{x}_2)$

where:

$SE(\bar{x}_1 - \bar{x}_2) = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

(Use technology to find df; approximately $min(n_1 - 1, n_2 - 1)$ or more complex formula)

Common Mistakes

1. Confusing df: Always use df = n − 1, not n
2. Using s instead of SE: The standard error is $\frac{s}{\sqrt{n}}$, not just s
3. Wrong critical value: Look up $t^*$, not z, from the t-table
4. Forgetting conditions: Particularly the normality condition; state why it's satisfied

AP Exam Tip

Free-response questions often ask you to construct a t-interval. Show all steps: state formula, check conditions, identify $\bar{x}$, s, n, df, and $t^*$, calculate ME, and state the CI with interpretation. Partial credit is generous if you show correct understanding.