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

4.9.21 · Maths › Probability Theory & Statistics

Intuition Bada picture

Ye saare tests ek hi sawal poochhte hain: "Jo difference mujhe data mein dikh raha hai, kya wo real hai, ya sirf random noise ne akele yahi result de diya?" Hum ek test statistic banate hain = (jo signal hum measure karte hain) / (jo noise hum expect karte hain). Agar ye ratio kisi jaani-pehchaani reference distribution ki extreme tail mein jaake girta hai, toh sirf-noise-wali kahani naaqaabil-e-yakeen ho jaati hai aur hum null hypothesis $H_{0}$ ko reject kar dete hain. Ye chaar tests sirf is baat mein alag hain ki kya compare karte hain aur noise kaun si reference distribution follow karta hai:

z-test — ek mean, jab $σ$ known ho (ya bahut bada $n$ ho) → Normal.
t-test — ek mean, jab $σ$ sample se estimate kiya gaya ho → Student's $t$ .
chi-squared GOF — kya observed counts expected counts se match karte hain → $χ^{2}$ .
F-test — kya do variances equal hain (variances ka ratio) → $F$ .

1. Common machinery (WHY neeche sab kuch kaam karta hai)

WHY ratio? Ek raw difference jaise $\overset{x}{ˉ} - μ_{0} = 3$ ka koi matlab nahi jab tak tum na poochho "3 matlab kitne spread ke comparison mein?" Noise scale se divide karne par difference standardize ho jaata hai "number of standard errors" mein, jo alag-alag problems mein compare kiya ja sakta hai.

HOW Central Limit Theorem z/t tests ko power deta hai. Size $n$ ke sample mean ke liye: $E [\overset{ˉ}{X}] = μ, Var (\overset{ˉ}{X}) = \frac{σ ^{2}}{n}$

Variance ki derivation (first principles se): i.i.d. $X_{i}$ ke saath, $Var (\frac{1}{n} \sum X_{i}) = \frac{1}{n ^{2}} \sum Var (X_{i}) = \frac{1}{n ^{2}} n σ^{2} = \frac{σ ^{2}}{n} .$ Toh standard error hai $SE = σ / n$ . CLT se, $\overset{ˉ}{X} \approx N (μ, σ^{2} / n)$ , matlab $Z = \frac{X ˉ - μ _{0}}{σ / n} \sim N (0, 1) under H_{0} .$ Ye hai z-statistic — aur ye baaki sabke liye template hai.

2. z-test (mean, $σ$ known)

WHY $σ$ known hona special hai: denominator ek fixed constant hai, isliye sirf $\overset{x}{ˉ}$ mein hi randomness hai, jo exactly Normal hai — koi extra uncertainty andar nahi aati.

3. t-test (mean, $σ$ unknown)

WHY $n - 1$ se divide karte hain (Bessel's correction)? Deviations true $μ$ se nahi, $\overset{x}{ˉ}$ se li jaati hain. Sample apne mean ke "sabse kareeb" hota hai, isliye $\sum (x_{i} - \overset{x}{ˉ})^{2}$ systematically spread ko kam estimate karta hai. Hum 1 degree of freedom kho dete hain (constraint $\sum (x_{i} - \overset{x}{ˉ}) = 0$ ki wajah se), aur $n - 1$ se divide karne par $s^{2}$ unbiased ho jaata hai: $E [s^{2}] = σ^{2}$ .

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

4. Chi-squared goodness of fit

WHY $E_{i}$ se divide karte hain (first principles): counts ke liye, ek cell ka count roughly Poisson hota hai mean $E_{i}$ ke saath, isliye uska variance bhi $\approx E_{i}$ hota hai. Tab $(O_{i} - E_{i}) / E_{i}$ ek standardized $z$ -jaisi quantity hai; $ν$ inhe square karke sum karne par $χ_{ν}^{2}$ distribution milti hai.

WHY $ν = k - 1$ (bina extra params ke): counts par constraint hai $\sum O_{i} = \sum E_{i} = n$ (total fixed), jo $k$ cells se 1 degree of freedom remove karta hai.

5. F-test (do variances compare karna)

WHY ye F hai: har $s^{2}$ (scale tak) ek $χ^{2} / ν$ hai. Do independent $χ^{2}$ ka ratio, dono apne d.f. se divide kiye gaye, by definition $F$ hota hai. Toh F-test hai "do chi-squareds ki race."

Common mistake Steel-manned traps

"Main z use karunga kyunki Normal simpler hai, chahe $n = 6$ ho aur $σ$ unknown ho." Sahi lagta hai: CLT, Normal har jagah. Fix: unknown $σ$ aur chhote $n$ ke saath, estimated $s$ uncertainty add karta hai → fatter-tailed $t$ . $2.78$ ki jagah $1.96$ use karna zyada baar reject karata hai.

"Chi-squared cells jahan $E_{i} = 2$ ho, theek hain." Sahi lagta hai: ye sirf arithmetic hai. Fix: $χ^{2}$ approximation ko $E_{i} ≳ 5$ chahiye; chhoti expected values statistic ko inflate karti hain. Categories pool karo.

"Bada p-value prove karta hai ki $H_{0}$ true hai." Sahi lagta hai: reject fail kiya = accept kar liya. Fix: tum sirf evidence dhundhne mein fail hue; evidence ka na hona ≠ absence ka evidence.

" $ν$ mein estimated parameters subtract karna bhool jaana." Sahi lagta hai: $ν = k - 1$ hamesha. Fix: har parameter jo tum usi data se fit karte ho, ek aur d.f. cost karta hai: $ν = k - 1 - m$ .

Recall Ek 12-saal ke bacche ko explain karo

Socho tum kuch measure karte ho aur wo thoda off lagta hai jo tumne expect kiya tha. Kya wo sach mein off hai, ya tumhe sirf bad luck mili? Tum ek "weirdness score" banate ho = kitna off ho ÷ kitna wobble normal hai. Agar score bahut bada hai, toh "sirf bad luck" believable nahi raha, isliye tum keh dete ho kuch real chal raha hai. z aur t scores check karte hain ki koi average off hai ya nahi (t tab use hota hai jab tumhe pata nahi ki cheezein kitni wobble karti hain, isliye ye zyada cautious hai). Chi-squared check karta hai ki tumhara categories ka tally (jaise die faces) jo hona chahiye usse match karta hai ya nahi. F check karta hai ki koi cheez doosre se zyada wobble karti hai ya nahi. Same trick, "luck kya kar sakti hai" ke alag-alag shapes.

Flashcards

Mean ke liye z-test vs t-test kab use karte ho?

z jab population

σ

known ho (ya

n

bahut bada ho); t jab

σ

unknown ho aur

s

se estimate kiya gaya ho.

t-distribution ki tails Normal se moti kyun hoti hain?

Kyunki denominator mein

s

khud

σ

ka ek noisy estimate hai, extra variability add karta hai;

ν = n - 1

se control hota hai.

One-sample z-statistic likhो.

Z = (\overset{x}{ˉ} - μ_{0}) / (σ / n) \sim N (0, 1)

under

H_{0}

Sample variance mein

n - 1

se kyun divide karte hain?

Deviations

μ

se nahi,

\overset{x}{ˉ}

se hain; constraint

\sum (x_{i} - \overset{x}{ˉ}) = 0

ek d.f. remove karta hai, aur

n - 1

s^{2}

ko unbiased banata hai.

Chi-squared GOF statistic aur uska d.f. likhо.

χ^{2} = \sum (O_{i} - E_{i})^{2} / E_{i}

ν = k - 1 - m

(

m

= estimated parameters).

Har chi-squared term ko

E_{i}

se kyun divide karte hain?

Counts ~Poisson hoti hain variance

\approx E_{i}

ke saath, isliye

(O_{i} - E_{i}) / E_{i}

square karne se pehle standardized hoti hai.

F-test kya compare karta hai aur uska statistic kya hai?

Do variances ki equality;

F = s_{1}^{2} / s_{2}^{2} \sim F_{n_{1} - 1, n_{2} - 1}

, bada variance upar.

Variances ka ratio F distribution kyun hota hai?

Har

s^{2}

(scaled)

χ^{2} / ν

hai; do independent

χ^{2} / ν

ka ratio by definition F hota hai.

Chi-squared GOF validity ke liye rule of thumb?

Har expected count

E_{i} ≳ 5

; warna categories pool karo.

Kya bada p-value

H_{0}

prove karta hai?

Nahi — tum sirf uske against evidence dhundhne mein fail hue; ye

H_{0}

confirm nahi karta.

Connections

Central Limit Theorem — z/t ke liye $\overset{ˉ}{X}$ ki Normality justify karta hai.
Normal Distribution — $t$ ka limiting case jab $ν \to \infty$ .
Student's t-distribution aur Chi-squared Distribution aur F-distribution — reference tails.
Degrees of Freedom — information ke independent pieces count karta hai.
Hypothesis Testing aur p-value aur Type I and Type II Errors.
ANOVA — F-test ko many group means tak generalize karta hai.
Bessel's Correction — $n - 1$ unbiasedness argument.