Confidence intervals — derivation for mean, proportion

4.9.19 · Maths › Probability Theory & Statistics

1. The raw material: the sampling distribution

WHY variance $n$ se shrink karta hai? Independent variables ke sum ka variance add hota hai: $Var (\sum X_{i}) = n σ^{2} ⟹ Var (\frac{1}{n} \sum X_{i}) = \frac{1}{n ^{2}} \cdot n σ^{2} = \frac{σ ^{2}}{n} .$ $\overset{ˉ}{X}$ ka standard deviation — jise standard error kehte hain — $σ / n$ hota hai.

HOW hum iska shape jaante hain? Central Limit Theorem kehta hai bade $n$ ke liye, $\overset{ˉ}{X} \approx N (μ, \frac{σ ^{2}}{n}) ⟺ Z = \frac{X ˉ - μ}{σ / n} \approx N (0, 1) .$

2. Deriving the CI for the mean (σ known)

Common $z_{α /2}$ : 90% → 1.645, 95% → 1.96, 99% → 2.576.

Figure — Confidence intervals — derivation for mean, proportion

3. σ unknown → the t-distribution

WHY $t$ pe switch karte hain? Jab $σ$ unknown hota hai toh hum sample SD $s$ plug in karte hain. Lekin $s$ khud random hai aur chote $n$ ke liye $σ$ ko underestimate karta hai, isliye statistic ke tails mote ho jaate hain. Exact distribution (normal data ke liye) Student's $t$ hai $n - 1$ degrees of freedom ke saath.

Jab $n \to \infty$ , $t_{n - 1} \to N (0, 1)$ , toh bade samples ke liye dono methods agree karte hain.

4. Deriving the CI for a proportion

WHY yeh sirf ek special case hai? Ek 0/1 (Bernoulli) variable ka mean $p$ aur variance $p (1 - p)$ hota hai. Toh $\overset{p}{^}$ Bernoulli ka sample mean hai: $E [\overset{p}{^}] = p, Var (\overset{p}{^}) = \frac{p ( 1 - p )}{n} .$ CLT se, bade $n$ ke liye $\overset{p}{^} \approx N (p, \frac{p ( 1 - p )}{n})$ .

5. Worked examples

6. Common mistakes (Steel-man + fix)

7. Active recall

Recall Quick self-test (dekhne se pehle answer karo)

"95% confidence" ka actually matlab kya hai?
Interval mein $σ / n$ kyun use hota hai, $σ$ nahi?
$z$ ki jagah $t$ kab use karte hain?
Bernoulli ka variance kya hai, aur yeh proportion SE kaise deta hai?

Answers: (1) Is tarah banaye gaye 95% intervals repeated sampling mein $μ$ ko capture karte hain. (2) Hum mean ko bound karte hain, jiska SE $σ / n$ hai. (3) Jab $σ$ unknown ho (khaaskar chota $n$ ). (4) $p (1 - p)$ ; SE $= p (1 - p) / n$ , $\overset{p}{^}$ se estimate hota hai.

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho tum ek bade bartan ke soup ke kuch spoons taste karte ho uski average saltiness guess karne ke liye. Tum bilkul sahi nahi hoge, isliye "soup exactly 5 salty hai" kehne ki jagah, tum kehte ho "mujhe pretty sure lagta hai yeh 4.5 aur 5.5 ke beech hai." Agar kai doston ne bhi apne apne spoons taste kiye aur har ek ne ek aisa range banaya, toh 100 mein se 95 doston ke ranges mein actual saltiness hogi. Woh range hi confidence interval hai, aur yeh tighter hota jaata hai jitne zyada spoons tum taste karo (bada $n$ ).

8. Connections

Central Limit Theorem — engine jo $\overset{ˉ}{X}$ ko normal banata hai.
Standard Error — $σ / n$ wala shrinking spread.
Student's t-distribution — estimated $σ$ ke liye mote tails.
Hypothesis Testing — CI ↔ test duality ( $μ_{0}$ CI ke bahar ho ⇔ level $α$ pe reject karo).
Bernoulli & Binomial Distributions — proportion variance ka basis.
Sample Size Determination — $n$ solve karne ke liye margin formula ko invert karo.

95% confidence interval ka actually matlab kya hota hai?

Kai repeated samples mein, is tarah se banaye gaye 95% intervals true parameter ko contain karte hain (yeh procedure ki property hai, ek interval ki nahi).

Sample mean ka standard error?

σ / n

(ya

s / n

jab σ unknown ho).

Var (\overset{ˉ}{X}) = σ^{2} / n

kyun?

Variance of independent sum

n σ^{2}

tak add hota hai;

n

se divide karne par variance

1/ n^{2}

se scale hota hai,

σ^{2} / n

milta hai.

σ known hone par mean ke liye CI?

\overset{ˉ}{X} \pm z_{α /2} σ / n

σ unknown hone par mean ke liye CI?

\overset{ˉ}{X} \pm t_{α /2, n - 1} s / n

z

ki jagah

t

kyun use karte hain?

s

σ

estimate hota hai aur uncertainty add hoti hai;

t_{n - 1}

ke tails mote hote hain,

n \to \infty

par normal ban jaata hai.

n - 1

degrees of freedom kyun?

s

compute karne mein ek constraint

\sum (X_{i} - \overset{ˉ}{X}) = 0

use hoti hai,

n - 1

free deviations bachte hain.

Bernoulli(p) ka variance?

p (1 - p)

Proportion ke liye CI (Wald)?

\overset{p}{^} \pm z_{α /2} \overset{p}{^} (1 - \overset{p}{^}) / n

Proportion CI ke liye validity rule?

n \overset{p}{^} \geq 5

aur

n (1 - \overset{p}{^}) \geq 5

α ko α/2 per tail mein kyun split karte hain?

Two-sided interval har tail mein equal probability chodta hai taaki central band exactly

1 - α

hold kare.

95% ke liye

z_{α /2}

1.96 (1.645 NAHI, jo one-sided ke liye hai).

Margin of error

n

ke saath kaise change hota hai?

1/ n

ki tarah shrink hota hai —

n

ko chaar guna karo toh margin half ho jaata hai.