Summa Week 2 Flashcards
(109 cards)
1
Q
What does statistical inference involve?
A
statements about probability (e.g. probably, likely…)
2
Q
What does the probability theory deduce?
A
it lets us deduce propositions about the likelihood of various outcomes, if certain conditions are TRUE
3
Q
How is the probability of event A happening denoted by?
A
italicized p (A)
4
Q
What is italicized p (A)?
A
the sum of the probability of all elementary outcomes of event A
5
Q
How large can probability be?
A
0 less than p which is less than or equal to 1
6
Q
What are three approaches to probability and statistical inference?
A
classical (analytical) approach
frequentist approach
subjective approach
7
Q
What is an experiment?
A
tbd
8
Q
What is the sample space?
A
the sample space of an experiment or random trial is the set of all possible outcomes or results of that experiment
9
Q
What is an event?
A
an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned.
10
Q
What is an elementary outcome or a simple event?
A
an atomic event, or an outcome of an experiment to which a probability is assigned
11
Q
What is a sample?
A
tbd
12
Q
What is simple random sampling?
A
a simple random sample is a subset of a statistical population in which each member of the subset has an equal probability of being chosen
13
Q
What are equally probable events?
A
outcomes of an experiment to which a probability is assigned that each have a chance of being chosen when randomly assigned
14
Q
What is sampling without replacement?
A
like selecting tarot cards for a Celtic Cross spread, the results of which increase the likelihood of other
15
Q
What is sampling with replacement?
A
tbd
16
Q
What is the classical (analytic) approach to probability theory?
A
it makes certain assumptions (such as equally likely, independence) about a situation
e.g. if a sampling experiment has N possible outcomes (all equally likely to occur), this method would assign a probability of 1/N to each outcome
rolling a die
rolling dice and summing 2 numbers on top
17
Q
What is an easy method to determine probability?
A
the tree diagram
e.g. tossing a coin and flipping it to record the outcome of H or T for each toss
Start - Head - Head-HH-p=.250
- Head-Tail-HT- p=.250
- Tail-Head-TH p=.250
- Tail-Tail-TT p=.250
18
Q
True or false: If all possible outcomes are equally likely, the probability of the occurrence of an event is equal to the proportion of all possible outcomes favouring the event.
A
True
19
Q
If we define the event as drawing a King from a deck of cards, what is the probability of the event of p(King)?
A
4/52
4 kings in a deck of 52 cards, therefore 4 out of 52
20
Q
If we define the event as drawing the Ace of Spaces from a deck of cards, what is the probability of the event p(Ace of Spades)?
A
1/52
21
Q
If we define the event as drawing a FAce card from a deck of cards, what is the probability of the event p(Face Card)=?
A
12/52 = 6/26 = 3/13
(there are 3 face cards per suit - Jack, Queen, King, and 4 suits per deck - spades, hearts, diamonds, club, therefore 3 x 4 = 12/52)
22
Q
What is the probability of getting AT LEAST one correct answer for three true and false questions (assuming equal chances that the answer to a problem is correct or wrong)?
A
p(at least one correct answer) = 7/8 Do a tree diagram T-T-T T-T-F T-F-T T-F-F F-F-F - only one out of 8 that can be completely wrong, therefore 7/8 F-F-T F-T-F F-T-T
If the question was what is the probability of getting at least one answer wrong, then it would be 7/8 as well
If the question was what is the probability of getting at least two correct, then it would be 4/8 T-T-T T-T-F F-T-T T-F-T.
It would be the same amount for the probability of getting at least two wrong T-F-F F-F-F F-F-T F-T-F
23
Q
What are the three relationships that can exist between two outcomes (probability)?
A
mutually exclusive (disjoint)
independent
conditional
24
Q
What is a mutually exclusive or disjoint event (probability)?
A
two events that cannot logically occur at the same time, or events A and B intersect at zero/have a 0% chance of ever happening
e.g. a coin is tossed twice (H or T)
Start - HH, TH, TT, HT
Events A (HH and Event B (TT) are disjoint, therefore both cannot happen at the same time, whereas Events A (HT) and Event B (TH) are NOT disjoint
25
What is the probability rule for mutually exclusive/disjoint events?
If A and B are mutually exclusive/disjoint events, then p(A or B) = p(A) + p(B)
e.g. Drawing a heart OR a 3 from a deck of cards
= 13 hearts (ace,2,3,4,5,6,7,8,9,10,j,q,k) + 4 3s cards, - 3 of hearts = 16/52
26
What is also known as the mutually exclusive event?
a disjoint event
27
What is also known as a disjoint event?
a mutually exclusive event
28
What is the or/addition rule for probability?
p(A or B) = p(A) + p(B) - p(A and B)
e.g. the prob. of drawing either a heart or a spade from a deck of cards = 26/52
= 13 hearts/52 cards + 13 spades/52 cards - NO heart/spade cards 0/52 cards
= 26 cards/52 cards
29
What is the independent relationship between multiple outcomes in probability?
If Event A occurs that does not affect that probability of Event B occurring, then it is independent.
e.g.a tossing two coins allows EQUAL chances of the second toss being either H or T, regardless of the first toss
= p(A and B) = p(A)*p(B)
30
What is the independent probability rule?
If A and B are independent, then p(A and B) = p(A) * p(B)
e.g. tossing a H and then a T is 1/4
= 1H/2sides * 1T/2sides = 1Hx1T/4 sides = 1/4 = .250
31
Can a disjoint/mutually exclusive event be independent?
No! If A and B cannot occur together (or are disjoint), then knowing that A occurs DOES change the probability that B occurs
deck of cards tends to be disjoint UNLESS each card is placed back into the deck; coins seem to be independent most of the time
32
What is the prob of obtaining three heads with a three-coin toss?
```
p(three heads) = independent, therefore uses * [p(A) * p(B) * p(C) = independent]
p(H) * p(H) * p(H) = 1/8
HHH -one out of 8, therefore 1/8
HHT
HTT
HTH
TTT
TTH
THH
THT
```
33
What is the permutation symbol?
!
it refers to taking the current number and MULTIPLYING it by the consecutively lower numbers until 1
e.g. 4! = 4*3*2*1=24
34
What is the permutation/number of possibilities for 4 letters?
```
Pr^N = N!/(N-r)!
P4^4= 4! (4-4)! = 4!/0! =
4*3*2*1/1 = 24
e.g. 24 possibilities
ABCD
ABDC
ACBD
ACDB
ADBC
ADCB
BACD
BADC
BCAD
BCDA
BDAC
BDCA
CABD
CADB
CBAD
CBDA
CDAB
CDBA
DABC
DACB
DBAC
DBCA
DCAB
DCBA
```
35
What does the permutation 0! signify?
1.
it is what it is
36
What is the permutations formula in probability?
```
Pr^N = N!/(N-r)!
e.g. permutations of letters A,B,C,D
N = 4
r =4
P4^4 = 4!/(4-4)! = 4*3*2*1/0! =
24/1 = 24
```
37
What are the combinations for ABCD?
```
Cr^N = N!/r!(N-r)!
C2^4 = 4!/2!(4-2)! = 4!/2!2! =
4*3*2*1/2*1*2*1 = 6
```
38
What is joint probability?
in N independent trials, suppose Na, Nb, Nab denote the number of times events A, B, and AB occur respectively. According to the frequency interpretation of probability, for large N
p(A) = Na/N
p(B) = Nb/N
p(A and B) = Nab/N
39
What is the probability that you will engage in unsafe sex and that your partner will have AIDS?
p(A and B) = Nab/N
40
What is conditional probability?
the probability of one outcome that is DEPENDENT on the occurrence of the other outcome
e.g. drawing cards and continuing to draw more WITHOUT replacing them
41
The probability of drawing 2 heart cards from a deck
p(B I A) = p(A and B) /p(A), or p(A and B) = p(A) * p(B I A)
42
What is the multiplication law of probability?
p(A and B) = p(A) * p(B I A)
43
What are some conditions of conditional probability?
The probability of an event A MUST often be modified after information is obtained as to whether or not a related event B has taken place
44
```
What is the probability of hypertension if overweight?
= hypertensive and overweight .1
=not hypertensive and overweight ..15
= total .25
A = hypertension (dependent)
B = overweight
```
```
A = hypertension
B = overweight
p(has hypertension given that overweight) = p(A I B)
= .1/.25
= .4
```
45
```
What is the probability of being overweight given one has hypertension?
A = overweight (dependent)
B = hypertension
= hypertensive and overweight is .1
= not hypertensive and overweight is .15
```
```
A = overweight
B = hypertension
p(A I B)
= .1/.2
= .5?
NEED TO CHECK
```
46
What is the frequentist approach for probability theory?
the limit of its relative frequency in a large number of trials. HOwever, the relative frequency of an experiment can be anything
e.g. drawing M&Ms from a bag and replacing them and repeating the same experiment has proportion closer and closer to the estimated probability, or "the limit"
47
What is considered the value of a proportion when used in a frequentist approach?
e.g. the limit for M&Ms to be picked out of a bag when replaced over repeated samples
the proportion that is estimated for the same that actual probabilities measured tend to move closer and closer to with repeated trials
48
What is the subjective approach in probability theory?
probability represents an individuals SUBJECTIVE belief in the likelihood of the occurrence of an event
e.g. I think that tomorrow will be a good day, based on what I know
49
True or false:
Although the particular definition that you or I prefer may be important to each of us none of the definitions will lead to essentially the same result in terms of hypothesis testing, the discussion of which runs through the rest of the book. (It should be said that those who favour subjective probabilities often disagree with the general hypothesis-testing orientation.)
False. The definitions will essentially have the SAME result, they just have different approaches to them that are more helpful for certain scenarios
50
What is an a priori probability?
the number of events classified as A over the total number of possible events
i.e. p(A) = # of events A/total # of possible events
51
What is the a priori probability of a head on one toss of an UNbiased coin?
.5
the number of events classified as heads or 1 divided by the total number of possible events of heads plus tails, or 2
= 1/2
= .5
52
What is the a posteriori probability?
the number of times A has occurred over the total number of possible events
= p(A) = # of times A has occurred/total # of possible events
53
If I toss any coin ten times and I get 4 heads, what is the a posteriori probability of heads?
.4
= p(A) = # of times A has occurred (# of heads to have occurred, or 4 divided by the total # of possible events or 10
= 4/10
=.4
54
What does hypothesis testing compare a posteriori probability with?
a priori probability
55
Who was concerned with integrating "prior knowledge" into calculations of probability?
Thomas Bayes, who created the Bayes Theorem or Bayesian approach
56
What is the Bayesian approach?
a probability theory that is concerned with integrating "prior knowledge" into calculations of probability
57
What formula is this of:?
p(A/B) = [p(B/A)p(A)]/[p(B/A)p(A) + p(B/A)p(A)]
the Bayesian approach
58
Does this formula show if the hypothesis is true or false?
H = not H
D = certain given data
p(H/D) = [p(D/H)p(H)]/p(D/H)p(H) + p(D/H)p(H)
True.
...?
59
What are discrete probability distributions?
the probability of a SPECIFIC outcome
60
If a variable can take on one of a relatively small number of possible values, what kind of variable is it assumed to be?
a DISCRETE variable
61
What are five-point scales or socio-economic status (multinominal probability distributions) examples of?
discrete probability distributions (they are exact and specific)
62
What are continuous probability distributions?
the probability of obtaining a value that falls within a specific interval
63
What types of distributions are these examples of?
normal probability distribution, student's t-distribution, chi-square distribution, F-distribution
continuous probability distributions
64
The rule that says that the probability of a series of outcomes occurring on successive trials is the product of their individual probabilities is the _______ rule
multiplication/joint/independent
65
The rule that says that the probability of one outcome or the other outcome occurring on a particular trials is the sum of their individual probabilities is the ________ rule
addition/mutually exclusive/disjoint
66
The and rule is to _________ and the or rule is to __
a) multiplication rule; addition rule
b) addition rule; multiplication rule
c) multiplication rule; multiplication rule
d) addition rule; addition rule.
multiplication;addition
67
The probability of rolling either a 2 or a 5 on one roll of a standard die is:
a) .34
b) .25
c) .50
d) .03
a
68
The probability of rolling a 2 followed by a 6 on a standard die is:
a) .34
b) .25
c) .50
d) .03
d
69
A(n) ______ scale is a scale in which objects or individuals are broken into categories that have no numerical properties
nominal
70
A(n) _____ scale is a scale in which the units of measurement between the numbers on the scale are all equal in size
interval
71
Measures of _____ are numbers intended to characterize an entire distribution
central tendency
72
The ______ is the middle score in a distribution after the scores have been arranged from highest to lowest or lowest to highest
median
73
When mean and median are the same, the distribution has to be ______
symmetrical
74
Measures of ___ are numbers that indicate how dispersed scores are around the mean of the distribution
variation
75
When we divide the squared deviation scores by N - 1 rather than by N, we are using the ____ of the population standard deviation
unbiased estimator / degrees of freedom
76
s represents the __ standard deviation and o represents the _ standard deviation
sample; population
77
A distribution in which the peak is to the left of the centre point and the tail extends towards the right is a __ skewed distribution
positively or right
78
On average, __ statistic has the same value as the population parameter
an unbiased sample, and a POINT ESTIMATE
79
Letter grade on a test is to the __ scale of measurement and height is to the __ scale of measurement
a. ordinal, ratio
b. ordinal, nominal
c. nominal, interval
d. interval, ratio
a. ordinal, ratio
80
Weight is to the ___ scale of measurement and political affiliation is to the __ scale of measurement
a. ratio, ordinal
b. ratio, noinal
c. interval, nominal
d. ordinal, ratio
b. ratio, nominal
81
Qualitative variable is to quantitative variable as ____ is to _____
a. categorical variable, numerical variable
b. numerical variable, categorical variable
c. bar graph, histogram
d. categorical variable and bar graph; numerical variable and histogram
d!!!!!!!!!!!!!!
82
```
Inferential statistics allow us to infer something about the ____ based on the _____
a sample, population
b. population, sample
c. sample, sample
d. population, population
```
b. population, sample
83
Which of the following is not true?
a. All scores in the distribution are used in the calculation of the range
b. the average deviation is a more sophisticated measure of variation than the range; however, it may not weight extreme scores adequately
c. The standard deviation is the most sophisticated measure of variation because all scores in the distribution are used and because it weights extreme scores adequately
d. None of the other alternatives is false
D?
84
```
If the shape of a frequency distribution is lopsided, with a long tail projecting longer to the left than to the right, how would the distribution be skewed?
a. Normally
B. Negatively
c. Positively
d. Average
```
b. Negatively
85
Calculate the mean for the following distribution:
| 1, 1, 2, 2, 4, 5, 8, 9, 10, 11, 11, 11
6.25
86
Calculate the median for the following distribution:
| 1, 1, 2, 2, 4, 5, 8, 9, 10, 11, 11, 11
6.5
87
Calculate the mode for the following distribution:
| 1, 1, 2, 2, 4, 5, 8, 9, 10, 11, 11, 11
11
88
Calculate the range for the following distribution: 2, 2, 3, 4, 5, 6, 7, 8, 8.
6
89
Calculate the standard deviation for the following distribution: 2, 2, 3, 4, 5, 6, 7, 8, 8.
2.4
90
____ is the study of likelihood and uncertainty
Probability
91
The rule that says that the probability of a series of outcomes occurring on successive trials is the product of their individual probabilities is the ____ rule
multiplication
92
The rule that says that the probability of one outcome or the other outcome occurring on a particular trial is the sum of their individual probabilities is the ___ rule
addition
93
The and rule is to _____ and the or rule is to ___
a. multiplication ;addition
b. addition; multiplication
c. multiplication; multiplication
d. addition; addition
a.
94
The probability of rolling either a 2 or a 5 on one roll of a standard die is:
a. .34
b. .25
c. .50
d. .03
a. .34
95
The probability of rolling a 2 followed by a 6 on a standard die is:
a. .34
b. .25
c. .50
d. .03
d. .03
96
Let's say Bill has an IQ of 145 and is 52 inches tall
- IQ in the population has a mean of 100 and a standard deviation of 15
Height in the population has a mean of 64 inches with a standard deviation of 4
How many standard deviations is Bill away from the average IQ?
145-100/15 = 3 sd away from the average IQ
97
Let's say Bill has an IQ of 145 and is 52 inches tall
- IQ in the population has a mean of 100 and a standard deviation of 15
Height in the population has a mean of 64 inches with a standard deviation of 4
What is the Z score of Bill's height? Or what is Bill's height in Z scores?
52-64/4 = -3, which is significant. This guy is short!
98
Consider two sections of statistics:
- Gurnsey's class has a mean of 80, and s of 5; Marcantoni's class has a mean of 70 and s of 5
- Student 1 gets 80 in Gurnsey's class; student 2 gets 75 in Marcantoni's class
Which student did better?
Student 2 did better than Student 1
80 - 80/5 = 0
75-70/5 = 1
99
A ___ test is used when s and m are known and the sample is 30 or larger.
Z-test
100
Which of the following is an assumption of the Z test?
a. The data should be ordinal or nominal
b. The population distribution of scores should be normal
c. The population mean (m) is known, but not the standard deviation (s)
d. The sample size is typically less than 30
b
101
What is an interval estimate?
the range of datapoints that are likely to be within the distribution of the population, and therefore if the sampling distribution is found within this range, then we are more apt to fail to reject/accept the null hypothesis, not being able to show a reasonable effect between the two distributions
102
If an interval estimate is within the critical value range of the confidence interval of a population, what can we assume?
that the test hypothesis is such that it is statistically significant and indicative of an effect within the set alpha (need to note whether .-5, etc. and whether 1 or 2-tailed)
103
What size must the ztest statistic be in order to conclude a statistically significant result?
over 1.96 or under -1.96
104
What size must the ztest statistic be in order to conclude a statistically INSIGNIFICANT result?
under 1.96 or above -1.96
105
What do I always have to remember the standard deviation is divided by in order to create a population parameter that is UNBIASED?
divide it by the unbiased estimator, or the degrees of freedom
106
A wide interval estimate means we have __ certainty of the mean
LESS
107
What is the coefficient of variance?
the interval estimate divided by the standard error
108
what does a statistically INsignificant result at alpha suggest?
samples 95% of the time capture the true population mean; the true population mean is within the confidence interval
109
If we reject the null hypothesis, we assume the data shows evidence to suggest that the ___ hypothesis is _____. Otherwise, we assume the ____ hypothesis is ____, unless shown otherwise.
the null hypothesis is false, otherwise we assume it is true
