Stat 155, Section 2, Last Time

Stat 155, Section 2, Last Time• Interpreted Midterm Results

• Variance of Random Variables– From Probability Table– Properties:

• Ignores shift• Multiples comes through squared• Sum when independent

• Sampling Distributions– Binomial Distribution

• Calculate Probs with BINOMDIST• Mean & Standard Deviation

Reading In Textbook

Approximate Reading for Today’s Material:

Pages 334-351, 358-369

Approximate Reading for Next Class:

Pages 382-396, 400-416

Binomial Distribution

Normal Approximation to the Binomial

Idea:

Bi(n,p) prob. histo. curve

So can approximate Binomial probs with

normal areas

)1(, pnpnpN

Normal Approx. to Binomial

Before modern software, this was a critical

issue, since Binomial Table (C in text) only

goes to n = 20.

Normal Approx made this possible…

Now still useful, since BINONDIST conks out

around n = 1000 (but political polls need

n ~ 2000-3000).


Visualization of Normal Approximation:http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg20.xls

Bi(100,0.3): Looks really good

Bi(20,0.5): Chunky, approx. a little weak

Bi(20,0.05): p too small for good approx.

Bi(20,0.95): p too big for good approx.

http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg20.xls


When is Normal Approximation “acceptable”?

Textbooks “Rule of Thumb” (there are others):

Need: np >= 10 and n(1-p) >= 10

Relate to above examples:

Bi(20,0.5): np = n(1-p) = 10, boundary case

Bi(20,0.05): np = 4 < 10, poor approx.

Bi(20,0.95): n(1-p) = 4 < 10, poor approx.


Nice Illustration:

Publisher’s Website

• Statistical Applets

• Normal Approx to Binomial

• Shows how rule of thumb makes sense

http://bcs.whfreeman.com/ips5e/


HW:

5.21

5.23

Normal Approx. to BinomialHW:C18: In a political poll of 2000, 1010 will vote

for A. To decide how “safe” it is to predict A will win:

a. Calculate P{X >= 1010}, for X ~ Bi(2000,1/2) (0.327)

(“could happen”, so not safe to predict)b. Recalculate, assuming 1050 will vote for A

(0.0127)(now have stronger evidence, will build on this)


A refinement: Continuity Correction

Idea:

• Binomial only takes on Integer Values

• Not really Continuous

• Better viewpoint (i.e. better approx.):

Replace points by “± ½ bars”

Continuity CorrectionGraphically:

Approx P{X=0}

by area of

bar above

(-0.5,0.5),

under curve. I.e. P{X = 0} = P{-0.5<X<0.5}

Continuity CorrectionMore examples:

3 4 5 6 7

}5.75.2{}73{

}5.75.3{}73{

}5.65.2{}73{

}5.65.3{}73{

XPXP

XPXP

XPXP

XPXP

Continuity Correction

Next time, consider replacing above e.g by

something with number lines….


Excel example:http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg20.xls

Part 3

For Bi(20,0.2):

• Continuity Correction much better

• About an order of magnitude in % error



Notes:

• Gives substantial improvement for small n

• But doesn’t matter much for large n

• Class rules:

– do when easy

– Skip when it adds complications…


HW C 19: For X ~ Bi(25,0.4)

a. Check this is on the boundary of “OK” for

using a Normal Approximation.

Find:

b. P{6 < X <= 11}

c. P{6 <= X < 11}

Using:


Using:

i. The naïve Normal approx.

(0.607,0.607)

ii. Normal approx. w/ cont. corr.

(0.653,0.548)

iii. The exact Binomial Distribution

(0.659,0.556 )

Binomial Distribution

Two Important “scales”:

i. Counts: X ~ Bi(n,p) (done above)

ii. Proportions (~ percentages):

, on [0,1] scale

often very natural,

e.g. political polls

nX

p ˆ

Binomial for Proportions

Relationship betweens means and SDs:

(“on average, expect” )

pnpnn X

Xnn

Xp 111ˆ

pp ˆ

npp

pnpnn X

Xn

p

)1()1(

112

22

21

2ˆ

npp

p

)1(ˆ


Normal Approx for Proportions:

(just uses above means and SDs)

npp

pNp)1(

,ˆ


HW:

5.25 work with both BINOMDIST and Normal

Approx.

(BINOMDIST: (a) 0.1495 (b) 0.0418)

And now for somethingcompletely different….

An interesting advertisement:

http://www.albinoblacksheep.com/flash/honda.php



Section 5.2: Distrib’n of Sample Means

Idea: Study Probability Structure of

• Based on

• Drawn independently

• From same distribution,

• Having Expected Value:

• And Standard Deviation:

n

iiXn

X1

1

nXX ,...,1

XEX

X

Expected Value of Sample Mean

How does relate to ?

Sample mean “has the same mean” as the original data.

XE X

n

nXX

XXn

X n 11

11

XXXX nnn

11

Variance of Sample Mean

Study “spread” (i.e. quantify variation) of

Variance of Sample mean “reduced by ”

X

22

21

2

11

1n

nXX

XXn

X n

222

222

111XXXX n

nnn

n1

S. D. of Sample Mean

Since Standard Deviation is square root of Variance,

Take square roots to get:

S. D. of Sample mean “reduced by ”

XX n 1

n1

Mean & S. D. of Sample Mean

Summary:

Averaging:

1. Gives same centerpoint

2. Reduces variation by factor of

Called “Law of Averages, Part I”

n1

Law of Averages, Part I

Some consequences (worth noting):

• To “double accuracy”, need 4 times as much data.

• For “10 times the accuracy”, need 100 times as much data.


Nice Illustration:



• Law of Large Numbers



HW: 5.29

Distribution of Sample Mean

Now know center and spread, what about “shape of distribution”?

Case 1: If are indep.

CAN SHOW:

(knew these, news is “mound shape”)

Thus work with NORMDIST & NORMINV

nXX ,,1 ,N

nNX

,~


Case 2: If are “almost anything”

STILL HAVE:

“approximately”

nXX ,,1

X

nN

,


Remarks:

• Mathematics: in terms of

• Called “Law of Averages, Part II”

• Also called “Central Limit Theorem”

• Gives sense in which Normal Distribution is in the center

• Hence name “Normal” (ostentatious?)

limn

Law of Averages, Part II

More Remarks:

• Thus we will work with NORMDIST & NORMINV a lot, for averages

• This is why Normal Dist’n is good model for many different populations

(any sum of small indep. Random pieces)

• Also explains Normal Approximation to the Binomial


Explained by Law of Averages. II, since:

For X ~ Binomial (n,p)

Can represent X as:

Where:

Thus X is an average (rescaled sum), so

Law of Averages gives Normal Dist’n

n

iiXX

1

itrialonS

itrialonFX i 1

0


Nice Java Demo:http://www.amstat.org/publications/jse/v6n3/applets/CLT.html

1 Dice (think n = 1): Average Dist’n is flat

2 Dice (n = 1): Average Dist’n is triangle

…

5 Dice (n = 5): Looks quite “mound shaped”

http://www.amstat.org/publications/jse/v6n3/applets/CLT.html


Another cool one:http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html

• Create U shaped distribut’n with mouse

• Simul. samples of size 2: non-Normal

• Size n = 5: more normal

• Size n = 10 or 25: mound shaped

http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html

http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html


Textbook version:



• Central Limit Theorem



Class Example:http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg21.xls

Shows:

• Even starting from non-normal shape,

• Averages become normal

• More so for more averaging

• SD smaller with more averaging ( )n1



HW: 5.35, 5.37, 5.39, 5.45


A statistics professor was describing

sampling theory to his class, explaining

how a sample can be studied and used

to generalize to a population.

One of the students in the back of the room

kept shaking his head.


"What's the matter?" asked the professor.

"I don't believe it," said the student, "why not study the whole population in the first place?"

The professor continued explaining the ideas of random and representative samples.

The student still shook his head.


The professor launched into the mechanics of

proportional stratified samples, randomized

cluster sampling, the standard error of the

mean, and the central limit theorem.

The student remained unconvinced saying, "Too

much theory, too risky, I couldn't trust just a

few numbers in place of ALL of them."


Attempting a more practical example, the professor then explained the scientific rigor and meticulous sample selection of the Nielsen television ratings which are used to determine how multiple millions of advertising dollars are spent.

The student remained unimpressed saying, "You mean that just a sample of a few thousand can tell us exactly what over 250 MILLION people are doing?"


Finally, the professor, somewhat disgruntled with the skepticism, replied,

"Well, the next time you go to the campus clinic and they want to do a blood test...tell them that's not good enough ...

tell them to TAKE IT ALL!!"

From: GARY C. RAMSEYER• http://www.ilstu.edu/~gcramsey/Gallery.html

Stat 155, Section 2, Last Time

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