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handy symbols

σ - sigma, std dev of a population

μ - mu, mean of a population

xbar - mean of a sample

n =3D sample (size of sample)

N =3D size of population

∑ - sigma, summation

·= - multiplication

σ μ n N �= � ·

Assignment Schedule

Week 1

04/06/06

P.9 #1-8(all)

#17-20(all)

P.17 #1-3(odd)

P.18 #17-21

P.27 #9-19(odd)

04/11/06

P.44 #1-7 (odd) #13

P.69#1-7 (odd) #18

P.87#1-7(odd)

Week 2

04/13/06

P.99 #1-9 odd

P.128 #1-19 (odd)

Week 3

04/18/06

P.137 #7-15 (odd);

P 146 #1-15 (odd)

04/20/06

P.153 #1-11 (odd);

P 169 #1-31= (odd)

Week 4

04/25/06

Exam#1

04/27/06

p.192 #1-15(odd)

p.203 #1-27(odd)

Week 5

05/02/06

p.210 #1-15(odd);p.216 #1-9(odd)

p.238 #1-39= (odd)

week 6

05/04/06

p.246 #1-11(odd);p.256#1-11(odd)

Class2notes

Thursday, April 06, 2006

6:00 PM

hMsauer@sccd.ctc.edu

Martin Sauer

5 books in on Monday 10-12

Need TI 83/84 <--better or TI89<= /span>

Review of Homework #1

Sample: estimate

Parameter: entire population value<= /span>

n =3D sample (size of sample)

N =3D parameter (size of population= )

I.e. in US population, take a sampl= e of 1500 (n) of parameter 300,000,000 (N)

x (x-bar) : mean (avg) of a sample

μ (mu) : mean (avg) of a population

Discreet: countable outcomes/items<= /span>

Continuous: (measurements) time, temparture, distance, volume… things that are not exactly known

p can represent

percentage
proportion
probability

p (hat) : point estimate, estimates= the true proportion

p : true proportion

Types of error

sampling error
non sampling error

derive from human mistakes
note that for this class, we will assu= me there is no non sampling error

sampling

error

.17

p(hat)=3D.19

.21

95% confidence interval

<= img src=3D"BUS210_files/image003.gif" width=3D595 height=3D383>

(frequency)

Histogram w/ discrete values

6�= � <---raw frequency

.26 .42 .32 <--relative frequ= ency

Data

0�= � 1 2

n=3D19

…shows # of houses w/ 0, 1 or 2

Also can do a histogram with range = of values (class)

frequency distribution : histogram = that shows classes (intervals) of data counts

relative frequency dist: histogram = that shows the percent of data that falls into classes (intervals)

cumulative frequency dist (0give): a graph that shows the cumulative total of each data count

relative frequency: percent<= /p>

raw frequency: raw numbers

0= give histogram: shows cumulative frequency

= .26 .68 1.00

0�= � 1 2

…shows # of houses w/ 1 or less, = or 2 or less children in it

Frequency Distributions
- write classes as midpoints
- determining the midpoint of a class: =3D (HighVal + LowVal)/2<= /span>

???frequency histogram???

frequency

1 1

hours spent studying

Measures of Central Tendency : valus that describe the central (middle) characteristics of data
- Mean : (avg or μ xbar= ) the sum of data entries (sample size) divided by the number of entries=
- Median : the middle point of a data set= , when the data is arranged in ascending order
- Mode : the data entry that occurs the m= ost number of times

Better measure of a central tendency: Median

Weighted Average:

x(bar) =3D Σ(w*x)/Σw

where w =3D weights (or frequency),= x =3D data

=3D 54*0 + 117*1 + 72*2 + 2*3 + 12*= 4 + 3*5/300

Deviations : number of steps from m= ean to a number

sum of left side deviations =3D sum= of right side deviations

s =3D standard deviation of a sample (statistic)

σ =3D standard deviation of a population (parameter)

s =3D Square roo= t of (Σ(x - x(bar))2/n-1)

with a set of values: 3, ,8 and 4

x(bar) =3D 5

s =3D 2.6

= =3D square root of ((1-5)2 + (2-5)2 + 12-5)2/3-1)

= =3D square root of (16 + 9 + 49 / 2)

= =3D square root of (74/2)

= =3D 6.08 =3D 6.1

p(hat) =3D x/n

where x=3D# of successes and n=3Dsa= mple size

117

Ch1Reading

Saturday, April 08, 2006

6:12 PM

Basic Definitions

Data :

observations (measurements, genders, responses) that have been collected

Statistics :

a collection of methods for

planning experiments
obtaining data
then

organizing
summarizing
presenting
analyzing
interpreting
drawing conclusions

population :

complete collection of all elements to= be studied, contains ALL subjects to be studied

census :

collection of data from every member o= f the pop

sample :

sub-collection of members selected fro= m a pop

parameter :

numerical measurement describing some characteristic of a population

statistic :

numerical measurement describing some characteristic of a sample

quantitative data :

numbers representing counts or measure= ments

qualitative data :

(also categorical, attribute) can be separated into different cate= gories that are distinguished by some non-numerical characteristic <= /span>

discrete data :

result when the number of possible val= ues is either a finite number or a countable number (0, 1, 2, etc)

number of eggs laid by chickens

continuous data= :

(also numerical) result from infinitely many possible values that correspond to some continuous scale that covers a range of vals w/o gaps<= /li>
- number of gallons of milk produced by= cows

Nominal level of measur= ement :

characterized by data that consist of = names, labels, or categories only, where no ordering scheme is possible

yes/no/undecided
colors
don't use these for calculations

Ordinal level of measur= ement :

data with this characteristic can be arranged in some order, but differences between data values are meaningless or cannot be quantified

course grades
ranks like "livability"

how much more "livable" is= NYC than Seattle?

Interval level of measu= rement :

ordinal level + the difference between= any two data values is meaningful, but there is not natural zero starting point where none of the quantity is present

body temperatures
years AD

Ratio level of measurem= ent :

interval level + there is a natural ze= ro starting point.

differences and ratios are both meani= ngful
weights
prices

ratio test :

take two quantities where one number is twice the other, if twice correctly describes the quantities.

"twice as heavy" =3D OK
"twice as hot" =3D not

#18 on page 11 …what the 5000 is = supposed to be representative of is unclear, although it implies "households&qu= ot; in general. Does this mean we can= assume that 5000 is the population?

same question for #19

Critical Thinking and Statistics

voluntary response sam= ple : (also self selected) respondents choose to be included

problems

responses are representative of the t= ype of people who are likely to participate , not the entire population
examples

internet polls
mail in polls
telephone call in polls ( ads ask yo= u to call in)

common error: assume that the polled population is representative of a larger population

small samples <= span style=3D'font-family:Verdana'>: [obvious]

problems

extremely small samples don't necessa= rily represent a larger pop
even apparently large samples may not= be representative, based on size relative to the population

graphs
- problems
pictographs : objects used to depict data

ex: moneybags, stacks of coins, army t= anks, barrels, houses, etc
problems

using 3d objects to represent 2d data= , i.e. doubling the side of a cube increases volume by 8x

percentages :

ex: saying that we have "improved problem X by 100%" implies that problem X is totally gone, but = may not mean that at all

loaded questions :

phrasing of a question changes peoples' answers

"too little money is spent on welfare" vs " "too little money is spent on assistan= ce to the poor"

order of questions :

ordering the same question can effect answers, for whatever reason

refusals :

the person who refuses to answer poll questions is different than those who do

sugging : selling under the guise of a poll
correlation vs causali= ty :

correlation : two variables are relate= d
but correlation does not imply causali= ty
ex: wealth/IQ, obesity/truck driving, health/bmw driving

self interest study :

who commissioned the study?

precise numbers :

deriving a precise number from a large number based on percentage, etc., gives the misleading impression th= at it is an exact number
ex: 103,215,027 households in the US

partial pictures :

"90% of our cars sold in the last= 10 years are still on the road"

deliberate distortions= :

sometimes people just lie!

Design of Experiments

experiment :

apply some treatment, and observe the effects on subjects

observational studies = :

observe and measure specific characteristics, but don't attempt to modify the subjects being stud= ied
cross sectional study= :

data are observed, measured and colle= cted at one time

retrospective study <= /span>:

data are collected from the past by g= oing back in time

examine records, interviews, etc

prospective study : (also longitudinal, cohort)

data are collected in the future from groups (cohorts) sharing common factors

observe subjects with a potentially causative factor over time

Confounding :

occurs when the experiments is not abl= e to distinguish between the effects of different factors

Placebo Effect = :

subject incorrectly believes he is rec= eiving a real treatment and reports an improvement in symptoms

Hawthorne Effect :

subject responds differently because t= hey are part of an experiment

Rosenthall Effect : (also experimenter)

when the experimenter unintentionally influences subjects through facial expression, voice tone, etc.

Blinding :

subject doesn't know if they are recei= ving treatment or placebo

Double Blind :

subjects and experimenters don't know = what they are getting or evaluating

Blocks :

groups of subjects with similar characteristics that might affect the outcome of the experiment

Completely Randomized Experimental Design :

put subjects into blocks randomly

Rigorously Controlled = Design :

subjects are carefully chosen so that = they are similar within a block

random sample <= span style=3D'font-family:Verdana'>:

simple random sample <= /span>:

if, within a population, every possible sample of the same size has the same chance of being chosen

systematic sampling :

select a starting point, then select e= very kth element in the population

convenience sampling <= /span>:

use results that are easy to get

stratified sampling :

divide group into two or more subgroups within which some characteristic is shared, then select a sample from each subgroup (stratum)
more consistent than random sample, us= ed to reduce variation in results

cluster sampling :

divide pop into sections (clusters), randomly select some sections, then use all of the members of the selected sections

sampling error = :

chance sampling fluctuations between a sample result and the true population result

non-sampling error :

data are incorrectly collected, record= ed, analyzed, biased samples, etc.

Ch2Reading

Monday, April 10, 2006

2:07 PM

Characteristics of Data

Center :

representative or average value repres= enting the middle of the data set

Variation :

measure of the amount that the data va= lues vary among themselves

Distribution :

nature of shape of the distribution of= the data
bell shaped, normal distribution
uniform
skewed

Outliers :

sample values that lie very far from t= he vast majority of other sample values

Time :

changing characteristics of the data o= ver time

Descriptive Statistics = :

summarize or describe the important characteristics of a set of data

Inferential Statistics = :

use sample data to make inferences/generalizations about a population

frequency distribution = :

lists data values along with their corresponding frequencies/counts

lower class limits :

low end of range for each class in a distribution

upper class limits :

high end of range for each class in a distribution

class boundaries :

1/2 the gap between class values, added= to either end of a class

class midpoints = :

midpoint of a class

class width :

difference between two consecutive lower class limits
class width ≈ (highest value) - (lowe= st value) / number of classes

relative frequency :

relative frequency =3D class frequency = / sum of all frequencies
basically shows frequency as a %=

cumulative frequency :

sum of the frequencies for that class a= nd all previous classes

histogram :

bar graph where horizontal scale repres= ents classes, vertical represents frequency values
http://www.treeplan.com/BetterHistogram= _20041117_1555.htm

frequency polygon :

line segments
points directly above class midpoints
extend left and right so graph begins /= ends at X axis

todo: draw freq polygon

Ogive :

line graph showing cumulative frequency=
class boundaries along x axis
begin w/ lower boundary of 1st class
end w/ upper boundary of last class

todo: draw freq ogive

dotplot:

dots plotted along a number line, 1 dot= per data value

todo: draw dotplot

stem and leaf plot:

dots plotted along a number line, 1 dot= per data value
other types

expanded stem and leaf plot

multiple entries for each stem=
leaves are broken into subgroups with= their stems

condensed stem and leaf plot

multiple stems per line=
leaf entries for different stems are separated by an *

pareto chart:

histogram w/ non numerical classes, usua= lly show in descending order

other charts:

pie, scatter, time-series, etc

Measures of Center

Measures of Center or Central Tendency :=

values that describe the central (middl= e) characteristics of data

Arithmetic Mean = :

average, mean, etc.
formula

sigma (x)/n
x =3D data values
n =3D number of data values

<= img src=3D"BUS210_files/image009.gif" width=3D612 height=3D363>

Median :

x-tilde [really tilde???]
middle value

Mode

most frequently occurring value<= /span>

Midrange

highest value + lowest value / 2=

Mean from a freq distribution

pretend that all sample vals = in a class are the class midpoint
multiply each midpoint x by i= t's frequency f
sum frequency… ∑f
sum frequency * x… ∑(f·x= )
xbar =3D ∑(f·x)/∑f

Weighted Mean

allows you to assign a weight to each v= alue
xbar =3D ∑(w·x)/∑w
where w =3D weight

Skewed data

skewed to the left … mean, median, mo= de
skewed to the right … mode, median, m= ean

x-xbar seems to not work when I do = it, my hand calcs come out way off from the ti83 excel. #1 pg 87

biased vs unbiased method of calcul= ating standard deviation …TI-84 appears to use unbiased (I think), where MS stu= dent graphic calc and excel use either, but default to different ones… what's = the dif? [<---we will used UNBIASED in class. done.]

unbiased, "n-1" method, TI-84/excel default =3D sample?

biased, "n" method, Stude= nt default =3D population?

Range :

range =3D highest value - lowest value<= /span>

Standard Deviation :

a measure of variation of all values fr= om the mean
usually positive
zero if all data values are the same #<= /span>
outliers effect it dramatically<= /span>
given in the same units as the original= data set
std dev of sample data
- s =3D sqrt(∑[x-xbar]2/n-1)
- s =3D sqrt(n∑[x2]-[∑x]2)/n{n-1})
std dev of pop data

σ =3D sqrt(∑[x-μ= ]2/N)

Variance

sample variance : sample std dev square= d

variance =3D σ2
unbiased estimate of population varian= ce

population variance : σ (pop std dev) squared
not given in the same units as the orig= inal data set

i.e. min2 ??? hours2????

Round off Rule

carry one more decimal place than in the original data

Coefficient of Variation
- helps to compare variation for values t= aken from different populations
- sample: CV =3D s/xbar · 100%
- population: CV =3D = σ/μ · 100%
Finding Standard Deviation from a Freque= ncy Distribution

s =3D sqrt(n[∑{f·x2}]-[∑{f·x}]2 /= n[n-1])

Range Rule of Thumb

s ≈ range/4

Interpreting a known value of the std deviation

min usual value ≈ (mean) - 2 * (std d= ev)
max usual value ≈ (mean) + 2 * (std d= ev)

Empirical Rule for Data with a Bell Shap= ed Curve

68 - 95 - 99.7
1st std dev : 34% either side (68% of p= op)
2nd std dev : 1st+13.5% either side (95= % of pop)
3rd std dev : 2nd+2.4% either side (99.= 7% of pop)
4th std dev : 3rd+.1% either side (99.8= % of pop)

Chebyshev's Theorem

proportion of any set of data within K = std deviations of the mean is at least 1-1/K2

Measures of Relative Standing

or… good ways to compare values f= rom different data sets.

Zscore :

#of standard deviations that a given va= lue x is above or below the mean
value less than mean: negative z score<= /span>
value greater than mean: positive z sco= re

population

sample

...where xbar =3D mean of sample, m= u =3D mean of pop, s =3D std dev of sample and small sigma =3D std dev of pop.<= /p>

Quartiles

divide sorted values into 4 equal parts=
Q1 - separates bottom 25% from to 75%
Q2 - separates bottom 50% from top 50% = (same as median)
Q3 - separates bottom 75% from top 25%<= /span>

Percentiles

percentile value of x =3D (# of vals &l= t; x) / (# of vals > x)
notation for percentiles<= /li>
- n =3D # values in data set
- k =3D percentile being used
- L =3D locator that give position of a = value (12th in the list)
- Pk =3D kth percentile (P25 is the 25th percentile)
conversion from Kth to corresponding Da= ta Value

sort data low to hi
L =3D (k/100)n
if L =3D whole #

Pk =3D LthValue + Lth+1Value/2=

else

round L up
Pk =3D Lth value

Interquartile Range (IQR) =3D Q3 - Q1
Semiquartile Range =3D (Q3-Q1)/2<= /span>
midquartile =3D (Q3+Q1)/2<= /li>
10-90 percentile range =3D P90-P10

Exploratory Data Analysis

Exploratory Data Analysis :

process of using statistical tools to investigate data sets in order to understand their important characteristics

graphs, measures of center, measures of variation

5-number summary :

minimum value
1st quartile
median (2nd quartile)
3rd quartile
maximum value

boxplot : [also box-and-whisker diagram]

x-axis from min to max val
box w/ lines crossing x-axis at Q1, Q2,= Q3

on pg 107, why does he say the boxp= lots are "skewed right"?

min Q1= Q2 Q3 <= /span>max

Boxplot

Class3Notes

Tuesday, April 11, 2006

5:58 PM

sort out basic definition list

entering lists, getting stats

stat >> 1:edit >>= enter
stat >> calc >> 1= -var stats >> L1,L2

results

xbar --> average
∑x --> sum of all vals
Sx --> standard deviation
σx --> sum of population [this is wrong]

outlier :

extreme value waaaay outside first 4 std deviations

what exactly is "normal" dist<= /span>
standard normal distribution

mean =3D 0
std dev =3D 1

Z-score

z=3Dx-μ/σ
how far is x above/below the mean, in u= nits of std deviations?
σ = =3D std dev of population <---write = martin an email about where you saw small sigma listed as the sum of the population
ex

solving zscore for a a test where you = get 73%, the mean was 63, and the std dev was 7
z=3D73-63/7 =3D 1.43
ie you are 1.43 std devs above the mea= n

Probability :

the likelihood that an event will occur=
all probabilities add up to 100%=

sample space :

all of the possible outcomes of an expe= riment
single die cast --> S =3D {1,2,3,4,5= ,6}
coin --> S=3D{H,T}
sum of all possible outcomes is always = equal to 1

experiment :

a process that generates well defined outcomes

classical probability [also theoretical] :

where the # of outcomes in the sample s= pace is known, and each outcome is equally likely to occur

empirical [also statistical, experimental, relative frequency]:
- frequency of outcomes measured by experimenting
subjective :

estimate probability by making an educa= ted guess
w don't do this

σ - sigma, std dev population

μ - mu, mean of a population

xbar - mean of a sample

n =3D sample (size of sample)

N =3D parameter (size of population= )

∑ - sigma, sum of sample

·= - multiplication

todo draw curves with different mea= ns, same dev;

Ch3Reading

Wednesday, April 12, 2006

9:51 PM

Probability

Fundamentals

event :

collection of results or outcomes of a procedure

simple event :

outcome/event that cannot be further = broken down into simpler components

sample space :

all possible simple events for a proc= edure

Procedure

Example of Event

Sample Space

roll 1 die<= /span>

5 [simple]<= /span>

{1,2,3,4,5,= 6}

roll 2 die<= /span>

7 [not simp= le]

{1-1,1-2,�= �6-6}

Probability Notations

a probability

A,B,C :

specific events

P(A) :

probability of event A occurring

Rule 1 - Relative Frequency Approximation of a Probability= :

P(A) =3D # of times A occurred / # of t= imes trial was repeated
provides an estimate
Law of Large #s :

as # of observations increases, estima= tes get closer to actual P(A)
does the law of large #s only pertain to Rule 1?

Rule 2 - Classical Approach to Probability
- P(A) =3D # of ways A can occur / # of d= ifferent simple events
- i.e. P(A) =3D s/n
- requires equally likely outcomes=
Rule 3 - Subject Probabilities

guess P(A) based on knowledge of circumstances

P(A) as a number

0 =3D not possible
1 =3D certain
0 ≤ P(A) ≤ 1

Complementary Events

Complement of event A is Abar -->
this is all outcomes where A does not o= ccur

Rounding and Probabilities=

either give the exact value or round to= 3 sig digits
no fractions

Odds

actual odds against :
- actual odds against A occurring are P(Abar)/P(A)
- express as: a:b where a and b are ints= w/ no common factors
actual odds in favor :
- actual odds in favor of event A are reciprocal of odds against
- i.e. a:b (against) is b:a (for)=
payoff odds :

ratio of net profit to the amount bet<= /span>
payoff odds of against event A =3D (net profit) / (amount bet)

why is #13 on pg 129 .0154 instead of .0156 or .016? why is #11 on pg 129 .0501 instead of .05?

BECA= USE: Zeroes placed before other digits are not significant; 0.046 has two signifi= cant digits. Pasted from <http://www= .physics.uoguelph.ca/tutorials/sig_fig/SIG_dig.htm>

Addition Rule

note: Class4Notes suf= fice for CH3.2, book notes are minimally captured.
Compound event :

any event combining two or more simple events

Addition Rule :

P(A or B) =3D P(A occurs or B occurs o= r they both)
P(A or B) =3D P(A) + P(B) - P(A and B)=
when finding P of A or B, find total n= umber of ways each can occur, but each is only counted once<= /li>

Disjoint Events :

mutually exclusive
Solving P(A or B)

if A and B are disjoint=

P(A or B)=3DP(A)+P(B)<= /li>

else

P(A or B) =3D P(A) + P(B) - P(A and = B)

#9a on pg 147 is 1024, not 1032, right?

Addition Rule

Class4Notes

Thursday, April 13, 2006

6:05 PM

uniform distribution (i.e. rolling = a die)

1�= � 2 3 4 5 6

Outliers

outlier > [Q3 +1.5(Q3-Q1)]

~ 5 std dev above mean

outlier < [Q1 - 1.5(Q3-Q1)]

= 68% 95% 99.7%

<= img src=3D"BUS210_files/image018.gif" width=3D560 height=3D263>

48.3 60.6 72.9

μ =3D 72.9

σ =3D 12.3

x =3D 48 (author's heart rate)

Probability and Statistics=

P(X<=3D48) -> P(Z<=3D-2.02)
=3D.0217 (from z score table)
this number may be referred to as probability, percentile, percentage, proportion or area=

statistic

parameter

xbar

�= �

phat

[need to underst= and why this table seems weird to me, and why the meaning of σ and mu keep seeming to switch around]

phat =3D x/n

where n=3D sample size, x #of successes=
phat =3D probability of a sample=
P =3D probability of population<= /span>

venn diagra= m

Probability

P(A) + P(Abar) =3D 1
if S =3D {1,2,3,4,5,6}
and A =3D {4}
then

Abar =3D {1,2,3,5,6}

Complement Rule<= /span>

P(A) + P(Abar) =3D 1 or P(A) =3D 1- P(A= bar)
rolling a die

P(4) + P(!4= ) =3D 1

P(!4) =3D 1= -P(4)

=3D1-1/6
=3D 5/6

2 penny example

samples space =3D {TT,TH,HT,HH}<= /span>
Probability of each outcome is 25%
P(no heads) =3D P(TT) =3D 1/4
P(at least 1 tails) =3D 1 - P(HH) =3D 3= /4

Intersection of events

intersection of events A and B

A =3D {4,5,6} =3D # greater than 3
B =3D {1,3,5} =3D odd #<= /li>
P(A) =3D 50%, P(B) =3D 50%
P(A and B) =3D P(5) =3D 1/6

Union of events (addition law)

P(A) =3D 50%
P(B) =3D 50%
P(A or B) =3D P(A) + P(B) - 1 * P(A and= B)
= =3D .50 + .50 - 1/6
= =3D 5/6

joint probability (Multiplication Rule)

P(A and B) P(A)*P(B|A)
Deck of cards example

P(K and Q) …first two picks
P(K) * (Q|K) <---probability of Q occurring given K has already occurred
P(K) =3D 4/52
P(Q|K) =3D 4/51
i.e. P =3D 4/52 * 4/51

Intuitive Multiplication Law

P(A and B)

if A and B are independent

P(A and B) =3D P(A) * P(B)

else

P(A and B) =3D P(A)*P(B|A)

Conditional Probabilities<= /li>
- the probability of an event occurring g= ive than another even has already occurred
- P(A|B) … Probability of A given that = B has already occurred
- P(Clubs|Hearts) …Probability of picki= ng up a clubs given the first card you picked up was a hearts
- P(A|B) =3D P(A and B)/P(B)
Independent Events

two events are independent if the occur= rence of one event does not effect the occurrence of another event.<= /span>
i.e. P(A|B) =3D P(A)
sampling w/o replacement<= /li>

Dependent Events

P(A|B)≠P(A)
sampling w/o replacement<= /li>

Contingency Tables [also matrix]

major

male=

female

total

accounting<= /span>

170

110

280<= /p>

finance

120

100

220<= /p>

marketing

160

230<= /p>

mgmt=

150

120

270<= /p>

total

600<= /p>

400<= /p>

1000=

N =3D 1000

P(male) =3D= .6

P(Finance) = =3D .22

P(Accountin= g and Female) =3D .11

P(Accountin= g|Female) =3D P(Accounting and Female)/P(Female) =3D 110/400

P(Male|Mark= eting) =3D 160/230

A�= � Abar

total sample space (S). P(S) =3D 1<= /span>

P(A) =3D P(Abar) =3D 1.00

Class5Notes

Tuesday, April 18, 2006

6:07 PM

Principles Of Counting

The Multiplication Law= [also Fundamental Counting Principle]<= /span>

(fill in the formula= here)
ex

Jim has 10 shifts and 8 ties, how many combos…
(10)(8) =3D 80

3 coins, determine # of outcomes
coin 1 =3D 2; coin 2 =3D 2; coin 3 = =3D 2
2 * 2 * 2 =3D 8

Factorials

n! =3D n(n-1)(n-2)…1
# of way to arrange n objects in order=
ex

3! =3D 3*2*1 =3D 6
4! =3D 4*3*2*1 =3D 24
5! =3D 5*4*3*2*1 =3D 120
note: 0! =3D 1
note: 1! =3D 1

Permutations

an ordered arrangement of r objects from n object= s
after you order each obj, it can't be = used again in the list
nPr =3D n!/(n-r)!
where n =3D total in population, r # of objects
ex

# of ways t= hat 12 horses can come in 1st and 2nd place

nPr =3D 12!/(12-2)!

=3D 12!/10!

=3D 12 * 11* 10 * 9 * 8 * 7 * 6 = * 5 * 4 * 3 * 2 * 1/10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

=3D 12 * 11* 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1/10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 *= 1

=3D 12 * 11 =3D 132

permutation button on T84 for 12P2

12 >> MATH >> PRB >> = nPr >> 2 >> ENTER

if in nPr , n =3D r, then nPr =3D n!

Combinations

the number of ways to choose r objs fro= m a group of n objs without regard to order<= /li>
- nCr =3D n!/((n-r)!*r!)
- where n =3D total in population, r # of objects
- 12 >> MATH >> PRB >>= nCr >> 2 >> ENTER

Multiplication

what is this???
figure out how these rules fit together, this is not clear

Class6Notes

Thursday, April 20, 2006

6:13 PM

Complement Rule

P(A)

Lottery example

pick 6 numbers from 69 numbers (1-69)
69C6 =3D 69!/((69-6!)*6!) =3D 119,877,472
probability w/ 1 ticket =3D 1/119,877,4= 72 =3D .000000008

"random firing" example=

what is the chance that 4 specific individuals will be selected from 32
32C4 =3D 35,960
P(selecting those 4) =3D 1/35,960 =3D 2= .78xE-5

"airline" example

#of different routes between 3 cities
3! =3D 6
this is # of lists of 3 cities from a l= ist of 3 cities, so: 3P3

combo lock example

0-49 are possible for each number withi= n 3 selected
50^3 =3D 125,000

Test1Review

Sunday, April 23, 2006

12:39 PM

Test Review

FYIs

20 questions mult choice, t/f
have to show work

Test Problems

definitions

population

complete collection of all elements (subjects) to be studied
mean of a sample of a classroom's age= is a STATISTIC
full classroom: population
ages of everyone in the class: popula= tion data

parameter

numerical measurement describing some characteristic of a population
i.e. info collected regarding the ent= ire population

sample

sub-collection of members selected fr= om a population

statistic

numerical measurement describing some characteristic of a sample
i.e. info based on a sample of a popu= lation

symbols

σ :
- sigma, std dev of a population=
s :

standard dev of a sample

μ :
- <= /span>mu, mean of a population
xbar :

mean of a sample=

size of population

sample (size of sample)=

∑ :

sigma, summation=

fyis

see pg 9,11

quantitative vs qualitative
random sample definitions=

simple random sample :

all samples of size n have an equal c= hance of being selected

random sample :

each individual member of a pop has e= qual chance of being selected

std dev

Definitions

standard deviation of a sample
statistic

σ : <= /li>
- standard deviation of a population <= /span>
- parameter
Formula

s =3D sqrt (Σ(x - xbar)^2/n-1)
x =3D sample, xbar =3D sample mean, = n =3D #of vals
explained

fyis

show work on test

histogram

draw a histogram based on data given
mode

most common val

median

middle value

mean

sum (x)/n

sum(weight*x)/sum(weights)=

get basic stats from TI84<= /li>
- use stat>>calc>>1-var stats>>L1,L2>enter
convert a histogram into a relative hist= ogram

each histogram bar's val is divided by = the total number of values giving a percentage

Z-scores

shows how far is x above/below the mean= , in units of std deviations
formulae

z=3Dx-xbar/s

where x =3D sample val, xbar =3D samp= le mean, s =3D sample std dev

z=3Dx-μ/σ

where x =3D sample val, = μ =3D population mean, σ =3D pop std dev

fyis

pg99 problems… calculating Z scores = and saying what they mean [practice]
won't need z tables
understand problem given in

σ =3D 15, μ =3D 268 and x =3D 308
give a statement about interpretation= of the data

empirical rule

68% w/in 1 std dev; 95% w/in 2 std dev;= 99.7% w/in 3 std dev

sample space

all of the possible outcomes of an expe= riment
ex

single die cast --> S =3D {1,2,3,4,= 5,6}
coin --> S=3D{H,T}

complement rule

P(A) =3D 1 - P(Abar)

Conditional Probabilities<= /li>
- P(B|A)
- independent events :
- dependent events :
Counting Principles

The Multiplication Law :

also Fundamental Counting Principle
There are a * b * c possible combinati= ons of a, b and c

Factorial :

# of way to arrange n objects in order=
n! =3D n(n-1)(n-2)…1
note: 0! =3D 1
note: 1! =3D 1
ex: # of different routes through 3 ci= ties =3D 3!, or 3*2*1

Permutation :

an arrangement of r objects from n obj= ects without replacement, where order matters
nPr =3D n!/(n-r)!
where n =3D total in population, r # of objects
ex

choose 4 tv shows from 27 available, = order matters.
nPr = =3D n!/(n-r)!

=3D 27!/(27-4)!

=3D 27!/23!

=3D 27*26*25*24 * 23!/23!

=3D 27*26*25*24

special case: permutations when some ite= ms are alike

n items with n1 alike and n2 alike ...nk alike
P=3Dn!/n1! n2!...nk!
ex

Combination :

the number of combinations of r objs fr= om a group of n objs where order doesn't matter
nCr =3D n!/((n-r)!*r!)
where n =3D total in population, r =3D = # of objects
ex

# of 3 member committees that can be selected from a 9 member group
nCr =3D n!/((n-r)!*r!)

=3D 9!/((9-3)!*3!)

=3D 9!/(6!*3!)

=3D 9*8*7*6!/6!*3!

=3D 9*8*7/3*2*1

Goofy Examples

you are buying a computer, and to put it together choose from

5 printers
3 CPUs
4 mice
3 modems
…how many different systems can you = make

you are shopping and want to buy soup

7 different types of soup, how many different sets of 3 cans can you have
7C3

Test1ResultsReview

Thursday, April 27, 2006

6:10 PM

Better ways to find median=

find out n, divide by two, and see wher= e that lands you in the continuum
also, "med" in var stats on t= 84 shows this
example

cars sales made

occurrences

total

Method 1 (n, n/2)

n =3D 89

n/2 =3D 42<= /span>

[0 cars] 1�= ��8th digit

[1 car] 9…31st digit

[2 cars] 32= …70th digit

[3 cars] 70= …89th digit

if n/2 =3D = 42, the 42nd digit falls between the 32nd and 70th digits, therefore the median is 2 (cars sold).

method 2 (TI84)
create lists, with # of cars as L1

STAT>>= ;CALC>>1:1-Var Stats>>2nd>>L1>>,>>2nd>>L2

Use the val= ue listed as "Med"

Class7Notes

Tuesday, April 25, 2006

7:06 PM

Random Variables

also RV
variables are random in the sense that = we don't know what the outcome is going to be
experimental outcome that generates exa= ctly one numerical value
Discrete Random Variable<= /li>
- the var is a countable number suck as 0,1,2,3,4
Continuous Random Variable

the var is an uncountable value, like a measurement

in this class, f(x) is the same thing as saying P(x)
Probability Distribution For A Discrete Random Variable

follow up on this

sales vol (= x)

# of occure= nces

0 cars

1 car

117<= /p>

2 cars

3 cars

4 cars

5 cars

total

300<= /p>

Discrete probability distribution

f(x)=

.18<= /p>

.39<= /p>

2.24=

.24<= /p>

.14<= /p>

.04<= /p>

.01<= /p>

total

1.00=

Expected Value :

Theoretical long run average
E(x)
mu =3D E(x) =3Dsum(x*f(x)) <---same as weighted average formula

Standard Deviation of a Discrete Probabi= lity Distribution

smallsigma =3D sqrt(sum[(x-mu)^2 * P(x)= ]) <---same as std dev, but us= ing probability
you could write smallsigma =3D sqrt(sum[f(x-mu)^2/N) , but f/N =3D P(x), so they just use this inste= ad… I think this is like using probability as a different form of weighting=
no s version of this

FYI std dev of a pop

σ =3D sqrt(∑[x-μ]= 2/N)
s =3D sqrt(∑[x-xbar]2/n-1)

Coin Flip experiment

# of heads

f(x)

.125=

.375=

.125=

total

1.00=

mu =3D E(x) =3D sum(xf(x))=
this gives us the most likely, or expect= ed value

Binomial Probability Distribution=

requirements of a binomial experiment

there are a fixed number of independent trials n
there are only 2 possible outcomes for= each trial, Success (S) or Failure (F)
the probabilities do not change for each trial

P(x) =3D [n!/x!(n-x)!] * px(1-p)= n-x

phat =3D proportion =3D successes/tria= ls =3D x/n
where

n =3D # of trials
p=3D probability of success on each t= rial
x =3D number of successes in n trials=
P(x) =3D probability of x successes i= n n trials

on ti= 2nd>>distr[vals]>>A:binompdf(n,p,x)=

binompdf means binomial probability distribution

see table A-1 of pg732

Mean and Std Dev

mu =3D E(x) =3D sum(x*f(x)) =3D n*p
smallsigma =3D sqrt[n*p(1-p)]

Ch4Reading

Thursday, April 27, 2006

10:16 AM

Probability Distributions

Random Variable :

has a single numerical value determined= by chance, for each outcome of a procedure
Subtypes

discrete random variable :

has a finite number of values, or val= ues that are discrete and countable, even though there may be an infini= te number of possible values

continuous random variable :

infinitely many values, associated wi= th measurements on a continuous scale in such a way that there are no = gaps or interruptions

Probability Distribution :

graph, table or formula that gives the probability for each value of the random variable
Probability Histogram...<= /li>

note the vals are in the middle of the b= ars, indicating that they are 1 u= nit wide. The area of these bars= are found by multiplying the probability val by the number.<= /li>
Probability Distribution Requirements:

∑P(x) =3D 1 AND 0<=3DP(x)<=3D1

Some Probability Distribution statistics=

mean [μ] =3D ∑[x·P(x)]
variance [σ2] =3D ∑[(x-μ)2·P(x)]
variance [σ2] =3D ∑[x2·P(x)]-μ2
std dev [σ] =3D √(∑[x2·P(x)]-μ2= )
round off rule :

carry one more decimal place than # of= dec places used for random var x

Range Rule Of Thumb

most vals should lie within 2 standard deviations of mean
max usual value =3D μ + 2σ
min usual value =3D μ - 2σ

Unusual Events and Probabilities <--check this

it is important to note that with proba= bility distributions, getting a value at or above a certain point, or at or below a certain point is more is the critical data we are looking for. For example, the proba= bility of getting exactly 10 girls (or 11, or 3, or 8) is very low and not significant to the probability distribution, but the probability of getting 10 or more is the data we are interested in.
unusually high: P(x) <=3D .05=
unusually low : P(x) <=3D .05=
Note: Martin is not particularly impres= sed with this line of thinking, he thinks it's "the opinion of the author"

he stressed the concept (in a test wit= h 0 to 10 possibilities) of…

P(X>=3D1= ) =3D 1 - P(X=3D0)

P(X>=3D1= ) =3D P(X=3D1) + P(X=3D2) + P(X=3D3) + P(X=3D4) + P(X=3D5) + P(X=3D6) + P(X=3D7) = + P(X=3D8) + P(X=3D9) + P(X=3D10)

Expected Value :

E =3D ∑[x·P(x)] =3D μ
i.e. the mean of a discrete random variable is the same as it's expected value

flip a coin 5 times:
- the mean # of heads is 2.5
- the expected value =3D 2.5
μ =3D xbar =3D E <--he said this will be on the test

Binomial Probability Distribution=

used with yes/no, boy/girl, acceptable/defection, etc.
requirements

fixed number of trials
trials are independent (one trial does= n’t affect the others)
outcomes must be classified into two categories only
probabilities for each trial remain co= nstant

Notation

P(x) =3D [n!/x!(n-x)!] * px(1-p)n-x
P(s) =3D p <--p=3D probability of a success
(F) =3D 1-p =3D q <--q =3D probability of a = failure
n =3D fixed number of trials
p =3D probability of success in = one of the n trials
x =3D specific number of successes in n= trials (any number between 0 and n)

note that binomial probability func o= n ti84 takes n,p,x

q =3D probability of failure in one of the n trials
P(x) =3D probability of getting exactl= y x successes among the n trials

Notes

x and p must refer to the same category being called a success
sampling is done without replacement
if sample size <=3D 5% of populatio= n size, events may be treated as independent
13b on 193 is a typical test question<= /span>

what does .000+ mean?... this means numbers that start with .000xyz, table A1 treats these as ~0

Class8Notes

Thursday, April 27, 2006

6:36 PM

Test Review

see Test1ResultsReview

Homework Review

see Ch4Reading

Review of Binomial Distribution

Random variable :

just the outcome of an experiment, may= be continuous or discrete

Binomial Experiment :

there are a fixed number of independent trials n
there are only two possible outcomes f= or each trial (S)uccess or (F)ailure
the probabilities do not change for ea= ch trial
P(S) =3D x/n

where

x =3D # of successes
n =3D number of trials=

expected value of a binomial experimen= t

μ =3D E(x) =3D∑xf(x) =3D np

where

n =3D number of trials
p =3D probability of success on eac= h trial
x =3D number of successes in n tria= ls
f(x) =3D probability of x successes= in n trials

it's important to bring books to class= on test nights, Table A1 will be used quite a bit from this point forwa= rd with Binomial Distribution problems
Example

What is your chance of getting to wor= k on time to work a certain number of days of the week?

n =3D 5 work days in a week
x is the # of days in the week you get t= o work on time
P(S) =3D .95 is the probability of being= on time on a given day
E(X) =3D μ =3D np

=3D 5(.95)<= /span>

=3D 4.75

= 0 = 1 2 3 4 5

0.000 0.0000 0.0011 .0214 .02036 ..7738

Binomial Distribution Test Problem

taking a 16 question mult choice exam w= ith 4 choices for each question.
n =3D 16
X =3D number of correct answers<= /span>
P(S) =3D .25
P(Passing with 75% or 12/16 right) =3D = P(12) + P(13) + P(14) + P(15) + P(16)

Poisson Probability Distribution :

A Poisson Distribution is a discrete probability distribution of a random variable that satisfies the following:

the interval can be time, area, distan= ce, or volume (continuous)
the experiment consists of counting the number of times, x, an event occurs in the interval
the probability of the event occurring= is the same for each interval
the events are independent regardless = of what might occur in another interval

examples

# of repairs needed in 10 miles of hig= hway (distance)
# of leaks in 100 miles of pipeline (distance/volume)
# of scratches in newly painted cars (= area)

Poisson Equation :

P(x) =3D μx·e-μ= /x!

where x =3D {0,1,2,…}=
μ =3D avg number of occurrences in the interva= l
e =3D 2.71828
x = =3D # of occurrences in the interval
P(x) =3D probability of x occurrences= in the interval

use e or ex button on TI84
or use Poissonpdf(= μ,x) in the same menu as the A:binompdf<= /span>
example

mean arrivals at a hospital between 6= -10 pm is 4 per hour what is the probability of 2 arrivals in an hour

P(2) =3D 42·e-4/2!

~10 people around the world are kille= d by sharks each year, what is the probability that exactly 3 are killed this year

P(3) =3D μx·e-μ/x! =3D 103(2.71828)-10/3!=3D.0076
doing this for 3 or more… this is = tough, because there are an infinite number of people. thus, use compliment rule to = solve this

P(X>=3D3) =3D 1-P(X<=3D2) …= because you can add up the probability of 0, 1, and 2

Means and Standard Deviations

Curves with different means, same stan= dard deviation

these have the same std dev, mean, = etc.

the middle one has a different std = dev than the other two

Review

empirical rule

68% w/in 1 std dev; 95% w/in 2 std dev; 99.7% w/in 3 std dev

Zscore

z =3D # of std deviations that x is fr= om the mean
z =3D (x-μ)/σ

actual vals 35 40 45 50 55 60 65

z scores -3 -2 -1 0 1 2 3

μ=3D 50

σ =3D 5

Problem: Find the probabili= ty of getting between 45 an 60 accidents in a 1 month per of time, given tha= t μ=3D 50, σ =3D 5 and this curve repre= sents accidents (use above diagram)

solution….

P(45 <= =3D x <=3D 60)

or...

P(-1 <+ z <=3D 2) =3D

=3D P(z <= ;=3D 2) - P(z<=3D-1)

=3D .9772 -.1587 (from table A2)

=3D .8385

alternatively, you can use a function from the TI84 distr menu. Use Normalcdf(L,U,μ,σ) where L =3D lower value, U =3D upper value.

in this example: Normalcdf(45,60,50,5)<= span style=3D'mso-spacerun:yes'> or Normalcdf(-1,2,0,1)

Auto Accidents in a 1 month period =

P(x) =3D ???

Created with Microsoft Office OneNo= te 2003
One place for all your notes

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