z-test, t-test, chi-squared goodness of fit, F-test: Visual walkthrough

4.9.21 · D2 · Maths › Probability Theory & Statistics › z-test, t-test, chi-squared goodness of fit, F-test

Step 1 — Ek measurement ek line par ek dot hai

YEH YAHAN SE KYUN SHURU KARTE HAIN. Har test "jo maine dekha" ko "jo claim kiya gaya tha" se compare karta hai. Dono sirf usi line par positions hain, isliye pehle yeh agree karna zaroori hai ki ek number ek jagah hai.

PICTURE. Neeche orange dot $990$ par baitha hai. Magenta tick $1000$ par hai — yeh hai claim — woh value jo manufacturer ne pakki di hai. Hum claim ko $μ_{0}$ (padho "mew-nought") kehte hain: " $0$ " ek label hai jiska matlab hai null hypothesis mein woh number, boring status-quo wali story Hypothesis Testing se. Symbols clearly samajhne ke liye: ek single reading $x$ likhi jaati hai; poore sample ka average $\overset{x}{ˉ}$ ("x-bar") likha jaata hai. Jab sirf ek reading ho, toh $\overset{x}{ˉ} = x$ — ek number ka average woh number hi hota hai. Neeche $990$ par dot dono hi hai.

Figure — z-test, t-test, chi-squared goodness of fit, F-test

Step 2 — "10" ka raw gap abhi bhi kuch matlab nahi rakhta

KYUN. Ek coin ke mass ke liye das gram ka fark bahut bada hai lekin ek insaan ke mass ke liye woh dikh bhi nahi paata. Size ka matlab sirf natural spread ke relative hota hai. Toh hume ek ruler chahiye jiska unit wobble hi ho.

PICTURE. Do worlds, same gap of $10$ . Upar: measurements barely scatter karti hain — $10$ ka gap bheed se bahut door dikhta hai, clearly weird. Neeche: measurements bahut zyada scatter karti hain — $10$ ka gap noise mein kho jaata hai, bilkul ordinary. Same gap, bilkul ulte conclusions.

Step 3 — Ek measurement ki wobble: $σ$

YEH SYMBOL KYUN. Hume ek aisa number chahiye jo summarize kare "individual readings kitni door bhatak jaati hain?" $σ$ exactly wahi hai — data ke same units mein measure hota hai (hours, ml).

PICTURE. Magenta bell $μ$ par centred hai — population ka true mean, jo claim ke under $μ_{0}$ ke barabar hota hai. Violet arrows ek $σ$ dono sides tak phailte hain. Roughly $68%$ single readings us band ke andar aati hain. $σ$ literally "usual" zone ki half-width hai.

Step 4 — Averaging wobble ko shrink karta hai: standard error

AVERAGE KYUN LETE HAIN. Ek single bulb freak ho sakta hai. $n = 64$ bulbs ka average lena individual flukes ko cancel karta hai: kuch zyada chalta hai, kuch kam, aur woh partly wash out ho jaate hain. Average ki baaki bachi wobble standard error hai, likha jaata hai $SE$ .

PICTURE. Same claim, teen bells. Sabse chodi (violet) ek reading ki hai: spread $σ$ . $4$ readings ka average lena (orange) spread ko half kar deta hai. $64$ ka average (magenta) ise aathguna shrink kar deta hai — ek tall, tight spike. Zyada data → true mean ki sharper knowledge.

Step 5 — Gap ko wobble ke units mein measure karo: z-statistic

DIVIDE KYUN KARTE HAIN. Division gap ko ruler-units mein re-express karta hai. $- 2$ ka result matlab hai claim se do poore SE-lengths left side mein. Yeh number ab kisi bhi problem mein comparable hai — hours, grams, votes — kyunki units cancel ho jaate hain. Aur yahan Central Limit Theorem apna kaam karta hai: woh promise karta hai ki $\overset{x}{ˉ}$ bell shape follow karta hai, isliye yeh ratio standard bell follow karta hai.

PICTURE. Standard bell (centre $0$ , ek SE per gridline). Orange arrow claim se hamare data tak step karta hai: exactly $2$ SE-steps left. Woh count, apne sign ke saath, Z hai.

Step 6 — "Too far" kahan hai? Rejection tails

$1.96$ KYUN. Standard bell ke saath, exactly $2.5%$ area $+ 1.96$ ke baad aur $2.5%$ $- 1.96$ ke baad hota hai — milke $5%$ . Woh $5%$ hamara tolerated false-alarm rate $α$ hai (Type I error). Area interpretation ke liye p-value dekho.

PICTURE. Bell jisme dono tails $\pm 1.96$ ke baad shaded hain (the rejection region). Humara $Z = - 2.0$ (orange) left shaded tail ke andar jaata hai — toh hum claim ko reject karte hain. Bulbs sach mein kam chalta hain.

Recall Khud check karo

Agar $Z = - 2.0$ hai aur cutoff $\pm 1.96$ hai, toh kya hum reject karte hain? ::: Haan — $∣ - 2.0∣ = 2.0 > 1.96$ , toh yeh tail mein jaata hai. Shaded area kya represent karta hai? ::: Agar claim sach hota toh itna extreme data dekhne ki probability — two-sided p-value threshold $α$ .

Step 7 — Jab ruler khud fuzzy ho: $σ \to s$ , aur $t$ ka janam

EK NAYI DISTRIBUTION KYUN. Step 5 mein denominator ek solid constant tha. Ab ratio ke upar aur neeche dono hil rahe hain. Ek fuzzy ruler ka matlab hai ki hum kabhi kabhi gap ko over- ya under-measure karte hain, Normal ke predict se zyada extreme values produce karte hain → fatter tails. Woh fatter-tailed curve hai Student's t-distribution, Degrees of Freedom $ν = n - 1$ se tune hoti hai. Jaise $n \to \infty$ , $s \to σ$ aur $t$ Normal mein wapis ghul jaata hai.

PICTURE. Do curves overlaid: crisp Normal (magenta) aur Student's $t$ few Degrees of Freedom ke saath (violet). Same centre, lekin $t$ beech mein zyada neeche dabha hua hai aur tails mein zyada heavy hai — isliye iska $95%$ cutoff aur door baitha hai ( $ν = 4$ ke liye $2.78$ , versus $1.96$ ). Fuzzy ruler ke baare mein cautious rehna goalposts ko peeche dhakelta hai.

Step 8 — Degenerate aur edge cases (koi gap mat chhodna)

Case A — bada $n$ . Jaise $n$ barta hai, $SE = σ / n \to 0$ : Step 4 ki bell ek spike mein collapse ho jaati hai. Ek chhota sa real gap bhi eventually ek giant $Z$ produce karta hai. Interpretation: kaafi data ke saath, koi bhi true difference detectable hai — lekin ek detectable difference real terms mein phir bhi trivially chhota ho sakta hai.

Case B — $\overset{x}{ˉ} = μ_{0}$ exactly. Gap $0$ hai, toh $Z = 0$ : bell ke dead centre par. Claim ke saath maximum agreement; hum kabhi reject nahi karte.

Case C — $σ \to 0$ (koi wobble nahi). Ruler zero tak shrink ho jaata hai, toh kisi bhi nonzero gap ke liye $Z \to \pm \infty$ . Interpretation: agar measurements sach mein kabhi wobble na karein, toh zaraa sa gap bhi infinitely surprising hai. Practice mein $σ$ kabhi exactly $0$ nahi hota.

Case D — t-test ke liye $n = 1$ (undefined!). T-formula $s$ se divide karta hai, aur $s^{2} = \frac{1}{n - 1} \sum (x_{i} - \overset{x}{ˉ})^{2}$ . $n = 1$ ke saath denominator $n - 1 = 0$ hota hai aur single point se koi deviations bhi nahi hoti — toh $s$ undefined hai aur $t$ compute nahi ho sakta. T-test ko $n > 1$ chahiye: tumhe kam se kam do readings chahiye pehle "kitna scatter karte hain?" ka koi answer bhi ho. (Z-test isse bach jaata hai kyunki woh ek given $σ$ use karta hai, na ki data se estimate kiya hua.)

Case E — mean se counts tak ( $χ^{2}$ ) aur counts se spreads tak ( $F$ ). Wahi ratio idea aage extend hota hai. Category counts ke liye hum har cell ko $(O_{i} - E_{i}) / E_{i}$ se standardize karte hain: yeh har count ko roughly Poisson maanta hai (variance $\approx E_{i}$ , toh standard deviation $\approx E_{i}$ ). Yeh ek approximation hai — ek large-sample / Poisson-normal limit jo tabhi hold karti hai jab har expected count reasonably bada ho (rule of thumb $E_{i} ≳ 5$ ); chhote $E_{i}$ ke saath yeh toot jaata hai aur tumhe categories pool karni padti hain. Us condition ke under, standardized cells ko square aur sum karne se Chi-squared Distribution milta hai. Do spreads compare karne ke liye hum do variances ka ratio lete hain, aur do chi-squareds ke beech ki race ANOVA mein use hone wali F-distribution hai.

PICTURE. Chaar mini-panels: (A) SE ka $n$ ke saath collapse; (B) $Z = 0$ peak par; (C) ratio blowing up jaise $σ \to 0$ ; (D) family tree $Z \to t \to χ^{2} \to F$ .

Ek-picture summary

Recall Feynman retelling — aise bolo jaise dil se ho

Maine kuch measure kiya aur woh jo unhone promise kiya tha usse $10$ neeche nikla. Lekin " $10$ " akela ek bluff hai — mujhe nahi pata yeh zyada hai ya kam. Toh main poochhta hun: mere readings ka average usually kitna hilta-dulta hai? Ek reading $σ$ se hilti hai, lekin maine unhe $64$ average kiya, aur averaging jiggle ko $64 = 8$ se calm karta hai, chhod ke $5$ ka jiggle. Ab main apna gap jiggles mein re-measure karta hun: $10 \div 5 = 2$ jiggles neeche. Do jiggles kaafi door hai — standard bell kehti hai itna door sirf badkismati se jaana $5%$ se kum time hota hai, toh main "bas badkismat tha" mein believe karna band kar deta hun aur kehta hun bulbs sach mein worse hain. Agar mujhe jiggle-size pata nahi hoti aur mujhe apne chhote sample se guess karna padta, toh mera ruler bhi shaky hota, toh main extra cautious rehta aur dare karne se pehle almost $2.8$ jiggles demand karta reject karne ke liye — woh cautious version hai $t$ -test (aur iske liye kam se kam do readings chahiye, warna koi jiggle measure hi nahi hota). Same trick categories count karti hai (chi-squared) aur do jiggles compare karti hai (F). Har test hai: kitna door, jiggles mein measured.