Foundations — Confidence intervals — derivation for mean, proportion

4.9.19 · D1 · Maths › Probability Theory & Statistics › Confidence intervals — derivation for mean, proportion

Yeh page kuch bhi assume nahi karta. Parent derivation ko touch karne se pehle, hum har woh letter, symbol, aur idea ka naam lete hain jo woh silently use karta hai, aur aisa order mein jahan ek cheez dusri par tikti hai.

0. Population vs. sample — do duniyaein

Neeche sab kuch do duniyaon mein se ek mein rehta hai. Dono ko confuse karna hi #1 tarika hai jisse log bhatak jaate hain, isliye hum pehle inhe draw karte hain.

1. $n$ — the sample size

Topic ko iske zaroorat kyun hai: is chapter ka har wobble $n$ badhne ke saath shrink hota hai. $n$ woh akela knob hai jo tum control karte ho. Bada $n$ → tighter interval. Yahi poori wajah hai kyun Sample Size Determination exist karta hai.

$n$ ki picture: yeh simply upar wali figure mein population blob se kitne dots khainche — bas itna hi hai.

2. $\mu$ — the population mean (the target)

Topic ko iske zaroorat kyun hai: $\mu$ woh cheez hai jise hum apne interval ke andar trap karna chahte hain. Yeh kabhi nahi hilta. Yahi fixedness ki wajah se "is interval mein $\mu$ hone ka 95% chance hai" — yeh galat hai — $\mu$ random cheez nahi hai; interval random hai.

Picture: number line par ek fixed pin. Humara kaam ek ring (interval) phenk kar hope karna hai ki woh pin ke upar land ho.

3. $\bar{X}$ — the sample mean (our estimate)

Abhi $\sum$ (capital Greek "sigma") ko decode karte hain, kyunki parent note ise baar baar use karta hai.

Topic ko $\bar{X}$ ki zaroorat kyun hai: yeh $\mu$ ke liye hamara andaza hai. Yeh ek fair andaza hai (yeh average par sahi hai), lekin kisi bhi ek sample ke liye thoda off rehta hai. Uss "thoda" ko measure karna hi poora game hai.

Picture: population blob se kuch dots lo, unka balance point dhundho — woh $\bar{X}$ hai. Alag mutthi bhar lo aur balance point shift ho jaata hai.

4. $\sigma$ and ${\sigma }^2$ — spread of one measurement

Square kyun karte hain phir square-root? Agar hum sirf signed distances ka average lete, toh plusses aur minuses cancel hokar zero ho jaate aur kuch nahi batate. Squaring signs ko khatam karta hai; final square root readable units restore karta hai.

Picture: individual bottles ke bell-shaped cloud ki width. $\sigma$ ek bottle ka wobble describe karta hai — not average ka wobble. Yahi distinction agle section ka crux hai.

5. $E[\cdot ]$ — the expectation operator

Isse pehle ki hum keh sakein sample mean "on average" kya karta hai, humein "on average" ko symbol ke roop mein define karna hoga, kyunki parent note iske upar nirbhar karta hai.

Topic ko iske zaroorat kyun hai: isse hum precisely keh sakte hain ki hamara estimate centred sahi hai. $E[\bar{X}]=\mu$ kehna matlab hai "agar countless samples lo aur unke saare $\bar{X}$'s average karo, toh exactly sach $\mu$ par land karoge" — yahi $\bar{X}$ ko ek unbiased andaza banata hai.

Picture: saare possible $\bar{X}$'s ke poore cloud ka balance point, jise hum aage draw karte hain.

6. The sampling distribution — show ka star

Yeh woh idea hai jis par poora topic pivots karta hai. Slowly padho.

Formulas se pehle, ek aur operator ko naam dete hain jo parent note bina ceremony ke use karta hai.

Spread kyun shrink hota hai? Jab tum $n$ independent measurements average karte ho, unke random highs aur lows partly cancel ho jaate hain. Averaging ek steadying act hai. Parent note ise algebraically prove karta hai; picture yeh hai: jitne zyada chamche soup taste karo, utna zyada tumhara average saltiness sach ki taraf settle hota jaata hai.

7. Standard error — woh wobble jo actually matter karta hai

Picture: upar wali figure mein, $\sigma$ wide pale cloud (individuals) ki width hai; SE narrow blue cloud (averages) ki width hai. Same centre, bahut alag widths.

8. The bell curve $\mathcal{N}$, the CLT, and its conditions

Bell curve aata hi kyun hai? Central Limit Theorem (CLT) ki wajah se: kaafi saari independent cheezein average karo toh bell mein pile up hoti hain, chahe originals ka shape kuch bhi ho. Yeh woh engine hai jo ek universal curve ko har cheez ke liye use karne deta hai — lekin yeh sirf tab fire karta hai jab uski conditions hold hon.

Topic ko conditions ki zaroorat kyun hai: agar yeh fail ho jaayein (jaise cherry-picked non-random data, ya wildly skewed population se tiny $n$), toh bell galat shape ki hai aur neeche har interval untrustworthy hai.

9. $\alpha$, $1-\alpha$, and the critical value $z_{\alpha /2}$

Handy trio: $90\%\to 1.645$, $95\%\to 1.96$, $99\%\to 2.576$.

10. $s$, degrees of freedom, and $t$ — when $\sigma$ is unknown

Kyunki $s$ khud $\sigma$ ka ek wobbly guess hai, iske saath standardise karna fatter tails wala curve deta hai — Student's t-distribution, likha jaata hai $t_{n-1}$. Fatter tails = wider interval = honest extra caution. Jaise $n\to \infty$, fat tails slim ho jaati hain aur $t_{n-1}\to \mathcal{N}(0,1)$.

11. $p$, $\hat{p}$ — proportions sirf 0s aur 1s ke averages hain

Isliye proportion standard error $\sqrt{p(1-p)/n}$, estimated as $\sqrt{\hat{p}(1-\hat{p})/n}$ — same "spread over $\sqrt{n}$" pattern.

12. Interval assemble karna — payoff

Ab har piece click karta hai. Parent note ke teeno formulas ek hi skeleton share karte hain:

Fill the skeleton teen tarah se:

Notice karo: same skeleton, teen costumes. Yahi woh "ek idea alag-alag costumes mein" hai jo upar promise kiya tha. Parent note har box ko algebraically derive karta hai; yahan tum dekh sakte ho ki yeh siblings hain.

Prerequisite map

Equipment checklist

Say each answer out loud before revealing.

Connections

• Central Limit Theorem — $\bar{X}$ ke liye bell curve kyun aata hai, aur uski conditions.
• Standard Error — $\sigma /\sqrt{n}$ wobble poori detail mein.
• Student's t-distribution — unknown $\sigma$ ke liye fat-tailed curve.
• Bernoulli & Binomial Distributions — proportions ke peeche 0/1 duniya.
• Sample Size Determination — knob $n$ jo har interval shrink karta hai.
• Hypothesis Testing — same $Z$ ruler ka mirror-image use.