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

4.9.20 · D2 · Maths › Probability Theory & Statistics › Hypothesis testing — null - alternative, test statistic, p-v

Poora plan ek nazar mein. Pehle padho, phir hum har box build karte hain.

Step 1 — Ek measurement ek noisy dart hai

KYA HAI. Hamare paas ek machine hai jo maani jaati hai ki har bottle mein kuch target amount bharta hai. Ek measured fill ko hum $X_1$ kehte hain. Doosra measure karo toh $X_2$ milta hai, aur aise $X_n$ tak, jahan $n$ bas "humne kitni bottles check kiyein" hai (ek simple counting number jaise $64$).

Har single measurement ek noisy dart hai: yahan tak ki ek perfect machine bhi kabhi exactly same amount do baar nahi daalta. Do invented words us noise ko describe karte hain:

YE DONO KYUN. Ek dartboard poori tarah summarised hota hai kahan cloud baitha hai ($\mu$) aur kitna chauda hai ($\sigma$). Noise ke baare mein aur kuch matter nahi karta jo aage aayega.

PICTURE. Neeche scattered pale-blue darts ka centre hai (yellow line $\mu$) aur spread hai (pink band $\pm \sigma$).

Step 2 — Bahut saare darts ko average karna wobble ko calm karta hai

KYA HAI. Ek noisy dart par trust karne ki jagah, hum unhe saare $n$ darts ko mila kar ek single number mein average karte hain — sample mean:

$\bar{X}=\underset{\text{share equally}}{\underbrace{\frac{1}{n}}}\underset{\text{add up all }n\text{ fills}}{\underbrace{\sum _{i=1}^n{X}_i}}.$

$\bar{X}$ ko "X-bar" padho. Bar ka matlab hai "averaged." Symbol ${\sum }_{i=1}^n{X}_i$ instructions ka ek stack hai: $i=1$ se shuru karo, $X_i$ add karo, $i$ ek se badhao, $i=n$ ke baad ruko — yaani $X_1+X_2+\cdots +X_n$.

AVERAGE KYUN. Ek dart jumpy hota hai; bahut darts ka average barely hilta hai, kyunki yahan ek high fill wahan ek low fill ko cancel kar deta hai. Averaging hamaara noise-reducer hai.

PICTURE. Left: single darts, wide scatter. Right: unka average $\bar{X}$, same centre se chipka hua ek tight cluster. Same centre, smaller spread — yahi poora point hai.

Step 3 — Exactly kitna calmer? Variance $n$ se shrink hota hai

KYA HAI. Ab hum measure karte hain ki averaging ne cheezein kitni calm kiyein. Do facts, dono earned:

$\mathbb{E}[\bar{X}]=\mu ,\mathrm{Var}(\bar{X})=\frac{{\sigma }^2}{n}.$

• $\mathbb{E}[\bar{X}]=\mu$: averages ka average same centre par baitha hai. Averaging target ko hilata nahi, bas use sharpen karta hai.
• $\mathrm{Var}(\bar{X})=\frac{{\sigma }^2}{n}$: $\bar{X}$ ka spread single-dart spread ${\sigma }^2$ hai jo $n$ se divided hai.

$\frac{1}{n}$ KYUN. Independent darts ke liye, sum ke variances add hote hain: $\mathrm{Var}(\sum X_i)=n{\sigma }^2$. Lekin $\bar{X}$ us sum ko constant $\frac{1}{n}$ se multiply karta hai, aur ek constant $c$ variance se squared bahar aata hai: $\mathrm{Var}(cY)=c^2\mathrm{Var}(Y)$. Toh

$\mathrm{Var}(\bar{X})=(\frac{1}{n}{)}^2\cdot n{\sigma }^2=\frac{{\sigma }^2}{n}.$

Constant ko square karne se aaya $\frac{1}{n^2}$, adding se aaye $n$ ko beat karta hai — net shrink $n$ se. Yahi Central Limit Theorem aur Standard Error ka beej hai.

PICTURE. $n=1,4,16$ ke liye teen bell shapes: same centre $\mu$, har ek adha chauda jab $n$ chaar guna hota hai (kyunki width $\propto 1/\sqrt{n}$).

Step 4 — Wobble ka naam do: standard error

KYA HAI. $\bar{X}$ ka original units mein spread apna naam deserve karta hai — standard error:

$\mathrm{SE}=\sqrt{\mathrm{Var}(\bar{X})}=\frac{\sigma }{\sqrt{n}}.$

• $\sigma$ = ek dart ka spread (millilitres).
• $\sqrt{n}$ = averaging ne use kitna tight kiya.
• $\mathrm{SE}$ = $\bar{X}$ ka spread (bhi millilitres mein) — hamaara yardstick.

SQUARE-ROOT KYUN. Variance squared units (ml${}^2$) mein rehta hai; ml mein mean se compare karne ke liye hume ml par wapas aane ke liye square root lena padta hai. Zyada ke liye Standard Error dekho.

PICTURE. Wide dart-cloud (spread $\sigma$) ke paas narrow average-cloud (spread $\mathrm{SE}$), ek labelled bracket ke saath jo $\mathrm{SE}=\sigma /\sqrt{n}$ dikhata hai.

Step 5 — Boring story assume karo, phir gap measure karo

KYA HAI. "Boring story" ek specific claimed centre hai, ${\mu }_0$ (mu-nought) — jaise "machine sach mein average $500$ ml daalta hai." Yeh null hypothesis $H_0:\mu ={\mu }_0$ hai. Hum ise sach maante hain aur jo humne dekha aur jo yeh predict karta hai uske beech ka gap dekhte hain:

$\text{raw gap}=\bar{X}-{\mu }_0.$

• $\bar{X}$ = jo humne measure kiya.
• ${\mu }_0$ = jo $H_0$ ne promise kiya tha.
• unka difference = boring story se kitna door hai (ml mein).

$H_0$ ASSUME KYUN. Hume measure karne ke liye ek fixed centre chahiye. Boring claim hume wo fixed pin deta hai; sab kuch usi ke relative judge hota hai. Hum yeh nahi keh rahe ki $H_0$ sach hai — hum pooch rahe hain "agar hota, toh yeh gap kitna surprising hai?"

PICTURE. $\bar{X}$ ki bell ${\mu }_0$ par centred (assumption), hamaara observed $\bar{X}$ side mein marked aur raw gap ek yellow arrow ke roop mein drawn.

Step 6 — Gap ko yardsticks mein measure karo: z-statistic

KYA HAI. "$3$ ml" ka raw gap tab tak meaningless hai jab tak hum nahi jaante ki $3$ ml zyada hai ya thoda. Toh hum gap ko yardstick $\mathrm{SE}$ se divide karte hain:

$\boxed{Z=\frac{\stackrel{\text{gap in ml}}{\overbrace{\bar{X}-{\mu }_0}}}{\underset{\text{one yardstick in ml}}{\underbrace{\sigma /\sqrt{n}}}}}$

• Numerator $\bar{X}-{\mu }_0$: raw gap.
• Denominator $\sigma /\sqrt{n}=\mathrm{SE}$: ek standard error.
• $Z$: data $H_0$ se kitne standard errors door baitha hai — ek pure number, koi units nahi.

DIVIDE KYUN. Kisi length ko same-unit yardstick se divide karna units cancel kar deta hai aur jawaab deta hai "mein kitne wobbles door hoon?" $Z=-3$ ka matlab hai "claim se teen standard errors neeche." Central Limit Theorem ki wajah se, yeh $Z$ standard normal $\mathcal{N}(0,1)$ follow karta hai — ek fixed bell jis se hum probabilities padh sakte hain. (Jab $\sigma$ unknown hai aur hum sample spread $s$ substitute karte hain, toh same recipe fatter-tailed Student t-distribution par $T$ deta hai.)

PICTURE. Standard normal bell ek ruler par jo standard errors $-3,-2,-1,0,1,2,3$ mein marked hai, $\bar{X}$ ka raw gap teen yardsticks step off karke $Z=-3$ par land karta hai.

Step 7 — Surprise padho: shaded tail = p-value

KYA HAI. P-value standard bell ke neeche hamare $Z$ ke bahar tail(s) mein shaded area hai:

$p_{\text{two-tailed}}=2P(Z\ge |z_{\text{obs}}|).$

• $|z_{\text{obs}}|$: hum kitna bahar gaye, sign ignore karke.
• $P(Z\ge \cdot )$: chance ki ek fair (null-true) world ek tail mein kam se kam itna extreme kuch produce kare.
• factor $2$: "$\ne$" tests ke liye, extreme ka matlab dono taraf hai, toh dono tails shade karo.

AREA KYUN, TWO TAILS KYUN. Area = probability. Kyunki $H_1:\mu \ne {\mu }_0$ ek suspiciously high fill ko utna hi damning maanta hai jitna ek suspiciously low fill ko, hum dono ends shade karte hain. Ek right-tailed test ($>$) ek end shade karta hai; ek left-tailed test ($<$) doosra. Tail data dekhne ke baad choose karna cheating hai — pehle se fix karo.

PICTURE. Bell ke dono tails $\pm 3$ ke baad pink shade ke saath; chhota shaded area hai $p\approx 0.0027$.

Step 8 — Edge aur degenerate cases (kabhi skip mat karo)

KYA / KYUN / PICTURE har us corner ke liye jo reader hit kar sakta hai:

• $Z=0$ exactly ($\bar{X}={\mu }_0$): data claim par bilkul baitha hai. $p$ jitna bada ho sakta hai utna bada hai ($p=1$ two-tailed). Zero surprise — kabhi reject mat karo.
• $n=1$: $\mathrm{SE}=\sigma /\sqrt{1}=\sigma$. Ek dart, koi averaging help nahi — sabse wide possible bell. Method phir bhi kaam karta hai, bas weak hai.
• $n\to \infty$: $\mathrm{SE}=\sigma /\sqrt{n}\to 0$. Bell ek spike mein collapse ho jaati hai; koi bhi nonzero gap ek giant $|Z|$ deta hai aur reject karta hai. Ek bada sample ek tiny, meaningless difference bhi flag karta hai — statistical significance practical importance nahi hai.
• $\sigma$ unknown: sample spread $s$ swap karo, Student t-distribution par $n-1$ degrees of freedom ke saath $T$ use karo — spread ke baare mein hamaari uncertainty ke liye fatter tails pay karne ke liye. Chhote $n$ ke liye yeh reject karna harder banata hai, jaisa hona chahiye.

Ek-picture summary

Sab kuch ek board par: darts → average → $n$ se shrink → yardstick $\mathrm{SE}$ → yardsticks count karke $Z$ tak → $p$ ke liye tail shade karo.

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