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

4.9.21 · D1 · Maths › Probability Theory & Statistics › z-test, t-test, chi-squared goodness of fit, F-test

Parent note ko aaram se padhne se pehle, tumhe har woh symbol apna banana hoga jo woh tumhare saath throw karta hai. Yeh page har ek symbol ko zero se build karta hai: uska matlab simple words mein, uski picture kaisi hai, aur kyun yeh topic uske bina ji nahi sakta. Upar se neeche padho — har block mein sirf wahi symbols use hote hain jo uske upar define ho chuke hain.

1. Data, samples, aur humble subscript

Number line pe dots ki ek row ki imagine karo — har dot ek $x_i$ hai, aur unki kul sankhya $n$ hai.

2. Average $\bar{x}$ aur true mean $\mu$

Bent-E symbol $\sum$ (capital Greek sigma) sirf yeh kehta hai ki "sab ko add kar do." Neeche $i=1$ aur upar $n$ matlab hai "$i$ ko $1$ se $n$ tak jaane do." Toh ${\sum }_{i=1}^n{x}_i$ sirf $x_1+x_2+\cdots +x_n$ ka shorthand hai — aur kuch nahi.

3. Spread: variance ${\sigma }^2$, standard deviation $\sigma$, aur unke sample cousins $s^2,s$

Ek middle kaafi nahi hota — humein yeh bhi jaanna hota hai ki data us middle ke around kitna wobbly hai.

Squared distances kyun, plain distances kyun nahi? Plain distances $x_i-\bar{x}$ cancel out ho jaate hain — mean ke upar wale neeche wale ko exactly khatam kar dete hain (unka sum hamesha zero hota hai). Squaring us cancellation ko rokta hai, aur yahi woh spread measure bhi nikla jis par saare reference distributions built hain.

$n$ ki jagah $n-1$ se kyun divide karte hain? Yeh Bessel's Correction hai. Hum distances $\bar{x}$ se measure karte hain, lekin $\bar{x}$ apne data ki taraf khicha hua hota hai, toh woh distances thodi choti nikaIti hain. Chote number $n-1$ se divide karna answer ko compensate karne ke liye thoda upar nudge karta hai. Parent note ko isi ki zaroorat tab padti hai jab $\sigma$ unknown ho.

4. Standard error $\mathrm{SE}=\sigma /\sqrt{n}$ — average ka wobble

Yahi woh pivot hai jo z-test aur t-test ko kaam karta hai.

$n$ nahi, $\sqrt{n}$ kyun? Averaging karne se errors partly cancel ho jaate hain, lekin randomly, toh shrinkage kitne average kiye uski square root ki tarah badhti hai, linearly nahi. Yeh ek sabse important number-fact hai jo topic use karta hai: apna data chaar guna karo, average ka wobble aadha ho jaata hai.

5. Bell curve: $\mathcal{N}(0,1)$ aur reference shapes ki family

Topic phir teen aur reference shapes se milta hai, har ek ek specific kaam ke liye bell ka ek distortion hai:

• Student's t-distribution — fatter tails wali ek bell, tab use hoti hai jab humein spread estimate karni padi ho.
• Chi-squared Distribution — counts ki squared surprises add karne ke liye ek right-skewed curve.
• F-distribution — do spreads ke ratio ke liye ek curve.

Tumhe abhi unke formulas ki zaroorat nahi — bas jaano ki har ek "luck akela kya karta hai?" curve hai.

6. Standardizing move: $Z$

Yeh ratio woh template hai jise parent note chaar baar repeat karta hai. Baaki sab sahi denominator aur sahi reference curve choose karne ke baare mein hai.

7. Greek letters aur Yes/No machinery

Prerequisite map

Equipment checklist

Khud ko test karo — daayni taraf cover karo aur zyaanoor se jawab do.