FAD1015 Confidence Interval Cookbook

Source: FAD1015 L21-L22 — Estimation of Population Mean. For exam application — follow the recipe.

Recipe 1: $\sigma$ Known

Use when the problem gives you $\sigma$ (population SD). Or when $n \ge 30$ and it gives $s$ — approximate.

Steps

# Action Formula
1 Find $\bar{x}$, $n$, $\sigma$, confidence level Read the question
2 Get $z_{\alpha/2}$ from table 90% → 1.645, 95% → 1.96, 99% → 2.576
3 Compute margin of error $E = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}$
4 Construct interval $\bar{x} \pm E$ → $(\text{LCL},; \text{UCL})$
5 Interpret "We are [X]% confident $\mu$ is between LCL and UCL"

Worked — Circuit Resistance

$n = 11$, $\bar{x} = 2.20\ \Omega$, $\sigma = 0.35\ \Omega$. 95% CI.

$$E = 1.96 \times \frac{0.35}{\sqrt{11}} = 0.207$$ $$95%\ \text{CI} = (2.20 - 0.207,; 2.20 + 0.207) = \boxed{(1.993,; 2.407)}$$

Recipe 2: $\sigma$ Unknown, Small $n$

Use when $\sigma$ is not given and $n < 30$. Must assume normal population.

Steps

# Action Formula
1 Find $\bar{x}$, $n$, $s$, confidence level Read the question
2 Compute $df = n - 1$ Degrees of freedom
3 Look up $t_{\alpha/2,,df}$ from $t$-table Row = $df$, Column = $\alpha/2$
4 Compute margin of error $E = t_{\alpha/2,,df} \times \frac{s}{\sqrt{n}}$
5 Construct interval $\bar{x} \pm E$ → $(\text{LCL},; \text{UCL})$

Worked — Random Sample

$n = 25$, $\bar{x} = 50$, $s = 8$. 95% CI.

$$df = 24,\quad t_{0.025,,24} = 2.064$$ $$E = 2.064 \times \frac{8}{\sqrt{25}} = 3.30$$ $$95%\ \text{CI} = 50 \pm 3.30 = \boxed{(46.70,; 53.30)}$$

Recipe 3: $\sigma$ Unknown, $n \ge 30$

Use when $\sigma$ is not given but $n \ge 30$. CLT lets you use $z$ with $s$.

Steps

Same as Recipe 1, but replace $\sigma$ with $s$.

# Action Formula
1 Find $\bar{x}$, $n$, $s$, confidence level Read the question
2 Get $z_{\alpha/2}$ from table 90% → 1.645, 95% → 1.96, 99% → 2.576
3 Compute margin of error $E = z_{\alpha/2} \times \frac{s}{\sqrt{n}}$
4 Construct interval $\bar{x} \pm E$ → $(\text{LCL},; \text{UCL})$

Worked — Tea Boxes

$n = 200$, $\bar{x} = 101.0$, $s = 2.78$. 99% CI.

$$E = 2.576 \times \frac{2.78}{\sqrt{200}} = 0.506$$ $$99%\ \text{CI} = 101.0 \pm 0.506 = \boxed{(100.494,; 101.506)}$$

Quick Pick — Which Recipe?

graph TD
    Q["Question says σ = ?"] -->|"Gives σ"| R1["Recipe 1<br/>Use z, formula with σ"]
    Q -->|"Does not give σ"| N["What is n?"]
    N -->|"n < 30"| R2["Recipe 2<br/>Use t, df = n-1"]
    N -->|"n ≥ 30"| R3["Recipe 3<br/>Use z, formula with s"]

Critical Values Cheat Sheet

z-Values (Standard Normal)

Test / CI $\alpha$ One-tailed $z_\alpha$ Two-tailed $z_{\alpha/2}$
90% CI 0.10 1.282 1.645
95% CI 0.05 1.645 1.960
99% CI 0.01 2.326 2.576

Memorize: 1.645 (90%), 1.96 (95%), 2.576 (99%).

t-Values (95% CI / Two-tailed $\alpha = 0.025$)

$df$ $t_{0.025}$ Typical $n$ When you see this...
5 2.571 $n = 6$ Very small sample
9 2.262 $n = 10$
14 2.145 $n = 15$
19 2.093 $n = 20$
24 2.064 $n = 25$ Textbook standard
29 2.045 $n = 30$ Threshold for CLT
40 2.021 $n = 41$
60 2.000 $n = 61$ Easy to recall: exactly 2
120 1.980 $n = 121$ Approaching $z$
$\infty$ 1.960 $n \to \infty$ Same as $z$

Key pattern: As $df$ grows, $t \to z$. At $df = 60$, $t = 2.000$ (exactly 2). At $df = \infty$, $t = z = 1.96$. For $df \ge 40$, the difference is negligible for exam purposes.

t-Values for All Three Confidence Levels

$df$ 90% CI ($t_{0.05}$) 95% CI ($t_{0.025}$) 99% CI ($t_{0.005}$)
1 6.314 12.706 63.657
2 2.920 4.303 9.925
3 2.353 3.182 5.841
5 2.015 2.571 4.032
10 1.812 2.228 3.169
15 1.753 2.131 2.947
20 1.725 2.086 2.845
25 1.708 2.060 2.787
30 1.697 2.042 2.750
40 1.684 2.021 2.704
60 1.671 2.000 2.660
120 1.658 1.980 2.617
$\infty$ 1.645 1.960 2.576

Common Exam Traps

Trap Fix
Using $z$ when $\sigma$ unknown and $n < 30$ Must use $t$ — $z$ gives too narrow an interval
Forgetting $n-1$ for $df$ $df$ is always $n-1$, never $n$
Wrong $\alpha$ column in $t$-table 95% CI → $\alpha = 0.05$ → use column $\alpha/2 = 0.025$
Saying "95% probability $\mu$ is in the interval" $\mu$ is fixed, not random. Say "95% confident"
Using $s$ instead of $\sigma/\sqrt{n}$ in margin formula The denominator is $\sqrt{n}$, not just $n$

Exam Shortcuts

  • 90% CI: $\bar{x} \pm 1.645 \times \text{SE}$
  • 95% CI: $\bar{x} \pm 1.96 \times \text{SE}$ (memorize: 1.96 ≈ 2)
  • 99% CI: $\bar{x} \pm 2.576 \times \text{SE}$
  • SE = $\sigma / \sqrt{n}$ or $s / \sqrt{n}$
  • If they ask "what changes when $n$ doubles?": width shrinks by $\sqrt{2}$ (about 30% narrower)

Related