MATH180A: Introduction to Probability

www.math.ucsd.edu/~ynemish/180a

This week:

homework #4 (due Friday, Nov 1, 11:59pm)

Today: ASV 4.3

Next: ASV 4.4

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( LT for the binomial

÷÷ïï÷ü÷ü"iRate of thumb :

For Are average ¥ npa - p ) > 10

p(q¥" ± ¥ -p±bY)#(b) -Pla)

Œ"NË:bËË?p;Ymmenaray

confidenceintervals.NO#tioh

Independent trials ,success rate p (unknown )

Sn = member of successes after n trials

Sn - Bon , p) . ( E. g. : ( biased ?) coin tossed n timesHo

LLN : SI → p , nao (in probabitity )If in is big , ¥ is close to p .

randommember

variable ← observable,estimant of p

Usually ,we don't know p , but we can get a

réalisation of § (flipping coin) for finit n .

M .

What can we soy about p ?

ConfidenceintervalssetupcLT : internal C-a. a) pr §← estimateur

pfqfmtT-ep-peaff-T.pt ) = 29cal - t

Pf tp -pk af¥ ) = Ca) - t = :XÛ confidence level

Questions : fixed (find a)1) For fixed n ,

find Est .

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PUP -pkE)et |2) For fixed E ,find ns.t .

PUÊ - pk E)Et l

Confidenceintervalsp( pt [ pr - e , pâte 3) ± y[ § - E , pr +E ) - y - confidence internal for p

§ is r - v . ⇒ internal is random .

= EnTata some réalisation of pr (a member) , say Ê*

[ ÎE , Ê# E) - y - confidence internal of p

Fat → est morte of p

ConfidenceintervalscomputationsoWhat with unknown pin formula ?

Pl In -pkavpa.pt.

P f) = 2 (a) - t = :p

N

Par - ne ÷}? :{fÏÏThe j - confidence Internal can

be taken as

[ § - E , ptite ] with• E >Ça forfixed n

• Tn > zaz for fixede

and 2 (a) - II y 20pct

cahidenceintervals.Exampfixednjfh.pecoin 10000 times .

Num best of Heads is 5370.

Compute a 99% - confidence Internal for p=P( Heads)M = 10000

, §* = 5370 = 0-537

201/2 - VIOOOT - E) - I I 0.99 0

(2- 100 - E) ? 0.995

2- IDO - E 22.58

{ I 2.58-200 ⇒ EZO - 0129

99% - confidence internat for p o ois given by [0.537 - 0.0129 , 0.537+0-0129]

Confidence internats . Example 2 ixed accuracy)

Flip a potentially biased coin . Hou mang times

should we repeat the expériment to be able to

compute a 95% - confidence internal for p = PCHeads)µ

of length 0.01 ?

j' = 0.95 ,E = 0.005

210fr -t - e) - t

= 2 (2- 0.005 - Fn ) - I I 0.95( O - 01 - Fn ) Z O - 975

0.01 . Fn ? 1.96 ⇒ nz 496)?- O

g- 95%- conf . Internal

I Repeat 1962 times, compute Ê* , [ pt - o.oot.pe +0.005)

for P

Confidenceintervalspollingwithoutgoinginto détails (see Example a -m)- Remark 4. 16

Part of population (unknown p) préfets product A- /supports candidate Bl. . .

We interview n individual s and K of them ogivepositive censurer (about product Kandi date) .

What can we soy about p ?

Fix confidence level j (often y = 0.95 )KAgain , pq is the estimât of p ,

[ ¥ - E , ¥ RE) gives 2 Kern ) - t - confidence infernal

⇒ 2 (« fn ) -pzz→ fixe d sample size→ fixedaccuracy

Confidentiellement rock vs rap )You ask 400 random1g chose n people living in SDif they prefer rock or rap . 230 reply that

they prefere rock music . Give a 99% confidence

internal for the part of the population that préfets rock .

• n = 400 , y = 0.99 , § = 24¥ = 0.575

2f(2E f407 ) - I I 0.99 ⇒ (40 E) I O - 995

=) 40 E I 2.58 E I = O.O 645

575% wittr[ 0.575 - O -0645

,0.575+0.0645 ) ← margin error 6.45%

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