Foundations — Hypothesis testing — null - alternative, test statistic, p-value, errors (Type I & II)

4.9.20 · D1 · Maths › Probability Theory & Statistics › Hypothesis testing — null - alternative, test statistic, p-v

Parent note padhne se pehle, tumhe har woh symbol apna banana hoga jo woh throw karta hai. Hum unhe ek ek karke build karte hain, bilkul zero se, har ek pichle waale pe tika hua. Koi bhi cheez use nahi hogi jab tak picture clear na ho.

1. Population vs sample — do duniyan

Poora subject isliye exist karta hai kyunki hum ek chhote sample ki taraf jhankne par majboor hain lekin uske peeche ki badi population ke baare mein kuch kehna chahte hain.

Figure dekho: blue cloud badi population hai; jo yellow dots humne scoop kiye woh sample hain. Do alag scoops do alag yellow patterns denge chahe blue cloud kabhi bhi na badle — woh wobble hi woh randomness hai jise hume control karna hai.

IS topic ko iski zaroorat kyun hai: har claim ("mean 500 ml hai") blue duniya ke baare mein hai, lekin har number jo hum compute karte hain yellow duniya se aata hai. Unke beech ka gap hi poora problem hai.

2. $n$ — sample size

Picture: dots gino. Wahi $n$ hai. Bada $n$ = zyada scoop kiya = tumhara yellow snapshot zyada trustworthy hai. Yeh akela number baad mein dono error rates ko quietly control karta hai, isliye iska dhyan rakho. Toh ab se, "a sample of $n$ measurements" ka matlab hai "$n$ yellow dots."

ABHI yeh flag kyun karo: neeche har result — average ka variance, Central Limit Theorem, poora standard-error machinery — silently i.i.d. par rely karta hai. Agar draws correlated hote ya alag alag clouds se aate, toh formulas toot jaate. Hum ise ek baar, loudly, yahan state karte hain.

3. Mean $\mu$ aur sample mean $\bar{X}$

Do flavours, aur inhe mix karna beginner ki sabse #1 galti hai:

$\bar{X}=\frac{1}{n}{\sum }_{i=1}^n{X}_i$

Toh $\bar{X}=\frac{1}{n}(X_1+\cdots +X_n)$ hai "har measurement jodo, barabar baanto." Yahi yellow balance point hai.

IS topic ko iski zaroorat kyun hai: $\mu$ woh cheez hai jiske baare mein null hypothesis ek claim karta hai; $\bar{X}$ woh evidence hai jo hum us claim ke against weighte hain.

4. Spread: variance ${\sigma }^2$ aur standard deviation $\sigma$

Balance point jaanna kaafi nahi — hume jaanna hai ki cloud kitna spread out hai. Ek tight cloud aur ek fat cloud same mean share kar sakti hain lekin bahut alag kahaniyan bata sakti hain.

Figure mein dono curves same $\mu$ par balance karte hain, lekin red wala fat hai ($\sigma$ bada — bahut spread) aur green wala skinny hai ($\sigma$ chhota — tightly bunched). $\sigma$ literally ek point ki mean se typical distance hai.

Square kyun karo phir square-root? KYA kiya humne: distances square kiye taaki pluses aur minuses ek fake zero mein cancel na ho sakein. WHY square root at the end: original units mein wapas aane ke liye taaki "$\sigma =8$ ml" meaningful ho. Yeh tool "wobble kitni wide hai?" sawaal ka jawab deta hai — spread ko sahi units mein koi aur nahi measure karta.

5. Normal distribution — the bell

Ek value ke upar curve ki height batati hai ki woh value kitni relatively common hai. Ek slice ke neeche area = us slice mein land karne ki probability. Total area = 1 (kuch toh hona chahiye).

IS topic ko iski zaroorat kyun hai: Central Limit Theorem (agla section) ki wajah se, sample mean $\bar{X}$ raw data ki shape parwah kiye bina is bell ko follow karta hai. Known shape matlab exact probabilities compute kar sakte hain — p-values ka raw fuel. Poori kahani ke liye dekho Normal Distribution.

6. Central Limit Theorem — bell free mein kyun appear hoti hai

Top panel: raw population ek lopsided, un-bell-like mess hai. Bottom panel: sample means ki distribution ek clean symmetric bell hai jo $\mu$ par centred hai. Yahi woh magic hai jo hume Normal-curve maths use karne deta hai chahe data khud Normal na ho. Deep dive: Central Limit Theorem.

IS topic ko iski zaroorat kyun hai: yeh $\bar{X}$ ko $z$ score mein standardize karne aur bell se probabilities padhne ka licence hai. Iske bina, parent ki derivation ka Step 3 illegal hai. (Licence tabhi valid hai jab draws i.i.d. hon — dekho §2.)

7. Standard error $\mathrm{SE}$ — average ka spread

Yahan ek subtle jump hai. Ek single measurement $\sigma$ se wobble karta hai. Lekin $n$ measurements ka average kam wobble karta hai, kyunki lucky-high aur lucky-low points partly cancel ho jaate hain.

Iska matlab: SE $\bar{X}$ ka apna standard deviation hai — agar tumne re-scoop kiya toh tumhara sample mean kitna jitter karta hai. $\sqrt{n}$ kyun, $n$ nahi: independent draws (§2) ke liye variances add hote hain, toh average ka variance $\mathrm{Var}(\bar{X})={\sigma }^2/n$ hai; SD wapas laane ke liye square root lena $\sigma /\sqrt{n}$ deta hai. Kaisa dikhta hai: sample chauguna karo aur average ki wobble aadhi ho jaati hai — diminishing returns. Poori picture Standard Error mein.

IS topic ko iski zaroorat kyun hai: SE test statistic ka ruler hai. Hum "claim se $\bar{X}$ kitni door hai" ml mein nahi, balki number of SEs mein measure karte hain — ek universal, unit-free scale.

8. Claim khud: $H_0$ aur null value ${\mu }_0$

"Claim se kitni door" measure karne se pehle, hume claim ke liye ek symbol chahiye.

Picture: ${\mu }_0$ number line par ek flag hai jahan boring duniya kehti hai true mean baith tha. Test jo kuch bhi karta hai woh hamare evidence $\bar{X}$ ki us flag se distance measure karna hai.

IS topic ko iski zaroorat kyun hai: $H_0$ woh defendant hai jise innocent mana jaata hai; ${\mu }_0$ woh precise story hai jiske against hum data test karte hain. Fixed ${\mu }_0$ ke bina koi "claim se distance" compute nahi hoti.

9. z-score — standard errors mein measured distance

Picture: bell ki horizontal axis ko "ml" se "SE ke steps" mein relabel karo. Ab $z=-3$ matlab hai "claimed mean se teen standard errors neeche" — kisi bhi problem mein, kisi bhi unit mein, impressively far. Wahi comparability hai jo precisely hum standardize kyun karte hain.

IS topic ko iski zaroorat kyun hai: test statistic $Z=\frac{\bar{X}-{\mu }_0}{\sigma /\sqrt{n}}$ precisely yahi z-score hai jo sample mean $\bar{X}$ par apply hota hai, claimed centre ${\mu }_0$ (§8) aur ruler SE (§7) use karte hue.

10. Jab $\sigma$ unknown ho: $s$, Bessel correction, aur t-statistic

Real life mein hum rarely population ka true $\sigma$ jaante hain. Toh hum ise sample se estimate karte hain aur estimate ko $s$ (sample standard deviation) kehte hain.

Ek guessed spread extra uncertainty add karta hai, toh bell ke thode fatter tails aa jaate hain. Woh fatter bell Student t-distribution hai, likha $t_{n-1}$. Zyada data ⇒ zyada degrees of freedom ⇒ t-bell wapas Normal ki taraf tighten hoti hai. Full treatment: Student t-distribution.

IS topic ko iski zaroorat kyun hai: yeh har us test ko power karta hai jahan $\sigma$ unknown ho (parent ke Worked Examples 2 aur 3).

11. Probability symbols: $P(\cdot )$, $\alpha$, $\beta$, aur $\mid$

Notation ka aakhri cluster — chhota lekin load-bearing.

p-value hai $P(\text{itna extreme data}\mid H_0\text{ true})$ — ise padho "evidence ki probability given boring world $H_0$ (§8)," kabhi nahi ulta. Bar ko flip karna woh prosecutor's fallacy hai jiske baare mein parent warn karta hai.

Prerequisite map

Har arrow kehta hai "target ko tab tak nahi samjh sakte jab tak source apna na ho." Sare raaste final box mein jaate hain — parent topic.

Equipment checklist

Self-test: kya tum reveal karne se pehle har ek aloud bol sakte ho?