Foundations — Hypothesis testing and p-values

1.3.20 · D1 · AI-ML › Probability & Statistics › Hypothesis testing and p-values

Isse pehle ki tum parent note ki ek bhi line par trust karo, tumhe use padhna aana chahiye. Yeh page har ek symbol aur idea jo parent use karta hai, unhe build-order mein list karta hai, taaki koi bhi notation tab tak na aaye jab tak tum uske malik na ho jao. Ek samajhdar 12-saal-ka bachcha bhi upar se neeche tak bina kisi jagah atak ke chal sakta hai.

0. Woh picture jis par hum baar baar wapas aate hain

Hypothesis testing ki almost har cheez ek bell-shaped curve par rehti hai. Abhi is picture ke saath comfortable ho jao aur baaki sab labelling hai.

Neeche sab kuch is curve ko draw karne aur usmein ek spot point karne ka tool hai.

1. Probability $P(\cdot )$ — "kitni baar" ki language

Picture: ek bade jar of outcomes ki soch lo. $P(\text{heads})$ jar ka woh fraction hai jo "heads" hai. Agar jar ka aadha heads hai, toh $P(\text{heads})=0.5$.

Topic ko iska kyun zaroorat hai: poora p-value ek probability hai — "kitni baar mujhe itna weird data dikhega?" Bina $P$ mein fluent hue tum p-value nahi padh sakte.

Bar $\mid$ (given that): $P(A\mid B)$ padhte hain "probability of $A$, given that $B$ pehle se true hai." Vertical bar ka matlab hai "assuming..."

2. Random variable $X$ — ek number jo chance se nikalta hai

Picture: ek slot machine. Tum lever khinchte ho (experiment chalate ho) aur ek number pop out hota hai. $X$ us number-generator ka naam hai; is baar jo value aata hai woh ek plain number hai jaise $65$.

Topic ko iska kyun zaroorat hai: test statistic (agle kuch sections mein) ek random variable hai. "65 heads kitna weird hai" poochne ke liye, tumhe imagine karna hoga ki $X$ kai values ho sakta tha, har ek ki koi na koi probability ke saath. Dekho 1.3.1-Random-variables-and-distributions.

3. Distribution — possible values aur unke chances ka poora menu

Picture: upar wala bar chart. Har bar heads ki ek possible sankhya hai; uunche bars = likely, chhote bars = rare. Fair coin ke bars bell shape banate hain jo Section 0 mein tha.

Do named distributions jo parent use karta hai:

Topic ko iska kyun zaroorat hai: p-value distribution ke neeche ek area hai. Koi distribution nahi, toh area measure karne ki koi jagah nahi.

4. Mean $\mu$ aur standard deviation $\sigma$ — center aur spread

Picture: $\mu$ wahan hai jahan curve peak karta hai (middle). $\sigma$ peak se woh horizontal distance hai jahan curve "steeply girna" se "flat hona" mein bend karta hai. Ek patli bell ka $\sigma$ chhota hota hai; ek moti bell ka $\sigma$ bada hota hai.

Topic ko iska kyun zaroorat hai: yeh kehne ke liye ki koi result "expected se dur" hai, tumhe measure karne ke liye ek center chahiye ($\mu$) aur measure karne ke liye ek ruler ($\sigma$).

5. Sample mean $\bar{X}$ vs. true mean ${\mu }_0$ — measured vs. claimed

Picture: number line par do dots. ${\mu }_0$ woh target hai jo null hypothesis paint karta hai. $\bar{X}$ wahan hai jahan tumhara actual result land kiya. Hypothesis testing unke beech gap measure karta hai.

Topic ko iska kyun zaroorat hai: har test statistic essentially $\text{(jo maine dekha)}-\text{(jo claim kiya gaya tha)}$ hai, yaani $\bar{X}-{\mu }_0$.

6. Test statistic — woh ek number jise hum curve ke against shade karte hain

Picture: tumhara saara messy data bell ki horizontal axis par ek dot mein crush ho jaata hai. Woh dot test statistic hai. P-value phir sirf us dot ke paar tail area hai.

Topic ko iska kyun zaroorat hai: tum curve ke against "tumhara data" shade nahi kar sakte — data ek single point nahi hota. Test statistic woh bridge hai jo ek poore dataset ko ek shadeable position mein convert karta hai.

7. Z-score $Z$ — "kitne rulers door?"

Worked reading: $Z=\frac{65-50}{5}=3$. Tum boring center se 3 standard deviations right mein ho — ek skinny tail mein gehre.

Saare cases (taaki tum kabhi surprise na ho):

• $Z=0$: result exactly expected par baitha hai. Bilkul boring.
• $Z>0$: result expected se upar hai (bell ka right side).
• $Z<0$: result expected se neeche hai (bell ka left side).
• $|Z|$ bada (jaise 3): ek tail mein kaafi bahar → surprising.
• $|Z|$ chhota (jaise 0.4): fat middle ke paas → unsurprising.

Bars $|\cdot |$ ka matlab absolute value hai — sign phenko, size rakho. $|-3|=3$. Hum iska use karte hain kyunki "3 steps left" aur "3 steps right" equally surprising hain; hume distance ki parwah hai, direction ki nahi.

Yeh standardizing trick 1.3.15-Central-limit-theorem aur 1.3.18-Confidence-intervals ka dil hai.

8. Tails, aur p-value ek area ke roop mein

Picture (upar): tumhara $Z$ ek spot mark karta hai. Us spot se aage sab kuch shade karo. Woh shaded area hi p-value hai.

9. Greek gatekeepers: $\alpha$, $\beta$, aur hypotheses

Picture: $\alpha$ bell ki tails par ek fence paint kiya hua hai. Agar tumhara p-value area $\alpha$ se chhota hai, tum fence ke bahar ho → $H_0$ reject karo.

10. Extra cheezein jo parent lean karta hai

Sab kuch topic ko kaise feed karta hai

Equipment checklist

Apne aap ko test karo — parent note padhne se pehle tumhe har jawab zor se bolne mein aana chahiye.