lecture 11

Question 1

why do we collect data in statistical studies

Answer 1

to learn about real world phenomena

Question 2

what is issue with statistical studies

Answer 2

can only collect finite sample of data
cannot tell us everything about system being studied
need to develop tools that let us talk about uncertainty associated with lack of complete information

Question 3

what does probability do

Answer 3

represents uncertainty and handles variability
describes nature of variability
how sample varies and what we can learn about population

Question 4

what do we recognize when we collect data

Answer 4

subject to variability =
measurement variability - using instrument to measure things
variability due to small changes in experimental condiitons

Question 5

give ex = repeated coin tosses

Answer 5

5 sequences of 10 tosses of fair coin
number of heads recorded is different for each sequence
outcome is variability
population= potentially infinite number of sequences of length 10
Probability helps describe and do calculations

Question 6

what is uncertainty

Answer 6

corresponds to a lack of perfect knowledge

Question 7

what could uncertainty be a consequence of

Answer 7

incomplete observation of system
unpredictable variation
simple lack of knowledge of state of nature (that we try to measure)

Question 8

describe - consequence of uncertainty - lack of knowledge of the state of nature

Answer 8

current state that is imperfectly observed
or
future state that is the result of a study yet to be carried out

Question 9

give ex: uncertain states of nature

Answer 9

outcome of single coin toss - outcome unknown since has not happened yet
milionth digit of pi - do not know off top of head
height of building i am in - fixed number but do not know right now
number of people in campus now
temp at noon tmr = have not observed it yet

Question 10

describe millionth digit of pi

Answer 10

fixed number but unless we know its value we are still in state of uncertainty when asked to asses it
we have lack of perfect knowledge

Question 11

give ex of uncertainty = coin tossing

Answer 11

2 possible results = heads or tails
outcome of toss uncertain before toss the coin, and after i toss the coin until i see the result

Question 12

give ex of uncertainty = thumbtack tossing

Answer 12

2 outcomes = point down or up
outcome uncertain before i toss thumbtack and after until i see resuslt

Question 13

what is probaility

Answer 13

numerical assessment of chance of a particular set of circumstances
chance of a particular event occurring given a particular set of circumstances

Question 14

describe term experiment

Answer 14

some situation that generates multiple possible outcomes
carry our an operation, make measurements etc

Question 15

what do we aim to do for an experiment

Answer 15

identify possible outcomes that could result
assign numerical values to collections of possible values to represent the chance that will coincide with actual outcome
lay out rules for how the numerical values can be manipulated

Question 16

basic concepts in set theory = set S

Answer 16

a set S is a collection of individual elements, such as S
we write s is one of the elements of S

Question 17

name what S may be

Answer 17

numbers
list of animals
closed intervals
any collection of possible outcomes
may be finite, countable or uncountable

Question 18

describe finite S

Answer 18

finite number of elements 1,2,3,4,5

Question 19

describe countable S

Answer 19

infinite number of elements that can be listed
1,2,3,4,5,6…..

Question 20

describe uncountable S

Answer 20

contains an infinite number of elements that cannot be listed
all values on continuum between 0 and 100

Question 21

describe A (subset of S)

Answer 21

A is a subset of S if it contains some, all or none of elements of S

Question 22

what is a empty set

Answer 22

subset that contains no elements

Question 23

what are 3 basic operations to manipulate sets

Answer 23

intersection
union
Complement

Question 24

what is the intersection

Answer 24

intersection of 2 sets A and B is the collection of elements that are elements of BOTH A and B
s is a ELEMENt of A intersection B
means that s is in A and s is in B

Question 25

for any A,B some or all in S

Answer 25

A intersection empty set = empty set
A intersection S= A
A intersection B = some or all in A
A intersection B = some or all in B

Question 26

what is union

Answer 26

union of 2 sets A B is set of distinct elements that are either in A or in B or in BOTh A and B
s is an element of A union B
means s is in A or s is in B or s is in the union of A and B

Question 27

for any A,B some or all in S

Answer 27

A union empty set = A
A union S = S
A some or all in A union B
B some or all in A union B

Question 28

describe complement

Answer 28

Complement of set A= collection of elements of S that are not elements of A
s is in A^C means s is in S but s NOT IN A

Question 29

we have for any A that… (complement)

Answer 29

A intersection A^c = empty set
A union A^c = S
(A^c)^c = A (elements not not in A are those elements in A)

Question 30

describe Extensions of intersections

Answer 30

(A ∩ B) ∩ c = A ∩ B ∩ C
ALL elements in A, B and C

Question 31

describe Extensions of unions

Answer 31

(A ∪ B) ∪ C = A ∪ B ∪ C
element of A,B or C, or any combo of 3 sets

Question 32

describe Extensions of intersections and unions

Answer 32

A ∪ (BC) = (A ∪ B) (A ∪ C)
in A or B and A or C
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
in A and B or A and C

Question 33

describe partitions and name types

Answer 33

we call A1,A2,…,Ak a partition of S if these sets are pairwise disjoint (mutually exclusive) or exhaustive

Question 34

describe partitions - pairwise disjoint

Answer 34

no overlap or intersection of pairs of elements
Aj ∩ Ak = empty set for all j not equal K

Question 35

describe partitions - exhaustive

Answer 35

when take union get S
union of all A
A’s cover whole of S but do not overlap
every s in S is an element of precisely one of Ak