PHILOSOPHY TEST 5

Statistical inference

drawing a conclusion about an entire population based on what you find in a sample taking from it

frequency of probability EXAMPLE

(P/A)=1/4 -means that if the experiment were repeated over and over, then as the # of trails > (increase to infinity) the frequency with which A will happen will converge on 25% of the time

Simple Random Sampling

-in which each object in the target population has the same probability of being included in our sample -can be done with or without replacement

Exhaustiveness

-A set of outcomes are jointly exhaustive of the possibility if NO OTHER OUTCOMES ARE POSSIBLE. -If you add up probabiliies, both has to equal 100% P(A v B)=100% - -Note that if A and B are jointly exhaustive then both of the following are true: P(A v B)= 1 P(A)+P(B)=1 (At least one or both will happen) -thought of as a sample space - All of the possiblities in combination must account for all of the space, which means their combined probabilities must sum to 100%

Systematic Error

-An error in testing the sample members -when it comes to testing a sample of people to find out THEIR OPINIONS - Usually result from questions that are 1. Poorly worded -Tendentious- worded to clear favor one response over the other -False alternatives- present respondents with too few options -Complex question- questions that ask at least two things at once, but to which respondents are only allowed to give one answer -Ambigious/vague cant put a clear meaning to it, nonsense question 2. Question Order 3. Circumstances (in which you ask question)

range of value

-HIGHEST value minus LOWEST value -helps us to get a sense of how widely the values in a list differ -just the range and average is not enough

Fallacy of Hasty Generalization

-The fallacy of drawing conclusions about the WHOLE population based on a sample too small to be considered a representative sample -When we reason about an entire population based on a sample that is too small 1. margin of error- the margin of error for a statistic is half the length of what is called the confidence interval 2. Confidence level 3. Variance

General Conjunction Rule FORMULA

-Used for "AND" and "BOTH" -Both occurring at the same time (ex. p.69) P(A&B)=P(A) x P(B I A) -P( B I A) are independent outcomes- if A and B are independent then P(B I A)= P(B) A and B independent (ex. p.69-70) P(B I A)= P(B)- the occurrence or non occurrence of A do not affect the probability of B. So, when A and B are independent we can simplify the rule to P(A&B)= P(A) x P(B) -Note: when drawing cards- draws are INDEPENDENT - Replacement/ you put back cards DEPENDENT- no replacement/ dont put cards back

Statistical inference EXAMPLE

-all eligible voters in the U.S. -All cars on the road in California -All cows in the United Kingdom -All stars in the Milky Way -47% of the people in my sample (drawn from the voting population) intend to vote for candidate Smith. So, 47% of the voting population intends to vote for candidate Smith.

Median

-middle number -high to low or low to high -when there's two in the middle add both number then divide by two -10, 22,48,57,112 median is 48

Contigent

-neither is certain or impossible. -each one will have a probability greater than zero and less than one -when two events are both contingent and independent of one another the probability that both will happen must be smaller than the probability of either event by itself -EX. 0<P(C)<1 - contingent

Gambler Fallacy

-the fallacy we commit when we let the outcomes in previous trails for some experiment influence our estimation of the probabilities in future trails when they both are INDEPENDENT -failure to distinguish two probabilities -try to imagine some plausible way in which the outcomes in previous trails COULD affect future outcomes

Disjunctive probability

-the probability that AT LEAST ONE of these events- OR BOTH occurs. -the v symbol is called a wedge, and it is used to write disjunctive (either-or) statements -P(A v B) is read "the probability of A or B"

conditional probability

-the probility that B will happen, given that A happens. -P(B/A) is read "the probability of B given A

Regression to the mean

-the tendency for an "outlier" (say, a score on an exam that is unusually low by the students own standards) to be followed by a value that is closer to the mean (the students average score on the exam so far)

mutually exclusive

-two events are mutually exclusive if the occurance of one precludes the occurance of the other-- ex. -THEY CANNOT BOTH HAPPEN TOGETHER. so, if A and B are mutually exclusive, then P(A&B)=0.

General Disjunction Rule (GDR)

-used to compute the probability that either or both of two events will occur P (A v B) = P(A) + P(B)- P(A&B) (if A& B are exclusive, then P(A&B)=0) (can eliminate P(A&B)) -the rule states that the probability of either A or B occurring is EQUAL to the probability of A added to the probability of B, minus the probability that both A and B occur -multiply the smaller number, add get greater probability - Because in some cases it will turn out that A and B can happen at the same time, and if we don't subtract P(A&B) WE'LL END UP COUNTING THIS POSSIBILITY TWICE

Mode

-value that's most often seen 1,2,2,3,4,4,4,5 4 is the mode

probaility examples

1. a. P(A v B)=1 - the probability of A or B b. P(A/B&C)= O.25 -probability that A/B&C will happen c. P(~B)=0 - B is difinit to happen d. 0<P(C)<1 - contingent e. P(A/B)=P(A)- B does not affect the probability of A 2. a. B is seventy-five percent probable P(B)= 75% (OR .75 OR 3/4) b. B is impossible P(B)=O.75 P(~B)= 1 3/4 c. It is certain that either A or its compliement will occur =1 P(A v ~A)=1 d. If B occurs, then A cannot occur P(A/B)=0 P(~A/B)=1 (both are equivalent) e. A and B cannot occur at the same time P(A&B)=0 P(A v B)=0 (A v B - means at least one will happen) (0- means niether happens) f. Niether A nor B can occur P (~A&~B)=1 g. It is probable as not that both A and B will occur P(A&B)= 1/2 h. A's occuring increases the probability that B will occur P(B/A)>P(B) 4) Give an example to show that P(A/B) and P(A&B) are two different things P (C/K)+1/4 P (C&K)= 1/52

Confidence Interval EXAMPLE

1. If my sample indicates that 20% of registered voters are unmarried and my margin of error is +5pts, then what is the confidence interval? 15-25%

Rules of probability -

1. probability of an even is a number between 0 and 1 2. 1 is CERTAIN to happen, then P(A)=1. 0 is am IMPOSSIBLE event, P(A)=0. 3. If A is neither necessary nor impossible- ex. it is CONTINGENT- then 0<P(A)<1 4. Negation Rule- P(A)=1-P(~A). That is, the probability that A happens is equal to one minus that probability that A does NOT happen. The event ~A is called the COMPLIMENT of A

General Disjunction Rule (GDR) EXAMPLE

Add together subtract overlap- avoid counting region twice 1. What is the probability of drawing either the ace of spades or a clib from a standard deck on a single draw? P(A) + P(C)= 1/52 + 13/52 = 14/52 = 7/26

Example Experiement

Also called a trail- every time you toss a coin its a trail Deck of cards tossing a die

Averages

Depending on the values, one type of average can be misleading (depending on numbers and figures)

Regression to the mean

Ex. Exam score Because the latest score was far from Marcia's average, her next score was likely to be closer to her average anyway, with or without the drug

Inverse Probability Fallacy EXAMPLE

Ex. lottery and roulette 1. The probability of getting a positive result if I have the disease is 99% (ex. P(+ I D)=99%) 2. So, given my positive result, it is 99% probable that I have the disease (ex. P(D I +)=99%) (consider: false poisitive rate, and how common is the disease)

Odds

Expressed differently than probabilities -Odds for or the odds against the outcome (unfavorable/favorable) -Uses a colon 19:1 -to the LEFT of the colon we record the number of UNFAVORABLE outcome that are possible, and to the RIGHT we record the number of FAVORABLE outcome that are possible -When the odds are for or against an outcome are 1:1, we can say the odds are even

Disjunctive probability EXAMPLE

If A is the outcome of getting a 1 when we roll a die, what is ~A? ~A= 2 v 3 v 5 v 6 (v means OR)

Inverse Gamblers Fallacy

P(A I B), there is an inverse probability P(B I A) -The inverse probailiy P(B I A) need not to equal to P(A I B) - When you think there must have been earlier trails that made outcome

classical of probability EXAMPLE

P(A)= 1/4 (occurs when tossing dice or drawing cards) -1/4 means that 4 things can happen, NONE is more likely than another and A happens in ONE of them

subjectivist of probability EXAMPLE

P(A)=1/4 -means that im 1/4 SURE A will happen/ have the same probability of being picked as anyone else

unconditional probability

P(A)=75% is read probability of A

General Conjunction Rule EXAMPLE

P. 193 1. An urn contains five red marbles, three green marbles, and two blue marbles. After selecting from the urn TWICE: A. Getting a red marble both times w/o replacement P(R1 & R2)= P(R1) x P(R2 I R1) = 5/10 x 4/9= 20/90= 2/9 B. Getting a blue marble on The first draw and green marble on second time w/o replacement P(B1 & G2)= P(B1) x P(G2 I B1) = 2/10 x 3/9= 6/90= 1/15 C. Getting a red marble on he first draw and blue marble on second time with replacement P(R1 & B2)= P(R1) x P(B2 I R1) = 5/10 x 2/10= 10/100= 1/10 D. Getting a blue marble both times with replacement P(B1 & B2)= P(B1) x P(B2 I B1) = 2/10 x 2/10= 4/100= 1/25 E. Getting a red marble with replacement at least once P(A)= 1-P (~A) 1- (NO RED) 1- (~R1 & ~R2) 1- [P (R1) x P (~R2/~R1)] 1- (5/10 x 5/10)= 75/100= 3/4 4. There is a 40% chance of rain in Seattle tomorrow, and a 30% chance of rain in Boston. Assuming that these events are independent, what is the probability it will rain in both places? 40% x 30%= 120% 5. Youre going to draw a card at random from a standard deck four times, and youre not going to replace any of the cards you draw. Which formula would provide the best figure for the probility of drawing an ace every time? c. 4/52 x 3/51 x 2/50 x 1/49 6. Which of the following (if either) is more probable? a. Chicago bears win the next Super Bowl b. Chicago bears win the next Super Bowl and it snows in Chicago at least once next year Neither is impossible- multiple together- has to be a smaller number A is correct

Averages EXAMPLE

P.209 example Employees make 100k/yr Misleading- average salary what they mean was MODAL AVERAGE -most often payed Assistant/ CEO misleading- should have been MEDIAN AVERAGE which is 25,000/yr P.210 1. 5, 20,45,50,50 mean- 34 median-45 mode-50 range- 45 2. What type of average is misleading when a data set has a few unusually high or low values? Skews mean up or down-makes mean misleading 3. What type of average is misleading when the most common value is the lowest and/or highest value? Mode 6. Sometimes instead of providing a mean average statisticians will provide a "trimmed mean" that drops some number of the highest and lowest values before the mean is computed. Why do this ? High don't skew it way up, lowest don't skew it way down

Negation Rule EXAMPLE

S P T (S-shirt, P-pants, T-tie) --- --- --- 4 3 2 How many ways to dress? Multiply (4*3*2)= 24 -probability of getting tails at least once when we toss a fair coin three times? P(tails at least once)= 1-P (No tails) = 1-P (All heads) = 1-P (H1,H2,H3) -Rule: P(A)= 1-P (~A) (both opposites/ compliment each other)

If A is independent of B, then B is independent of A

TRUE

If A, B, and C are jointly exhaustive of the possibilities, then 1-P(A v B v C)=0

TRUE

If A, B, and C are jointly exhaustive, then P(~A & ~B & ~C)=0

TRUE

If A, B, and C are jointly exhaustive, then P(A)+P(B)+P(C)=1

TRUE Have to add up all probability to equal 100%- only things that can happen

If A excluded B, then B excludes A

TRUE If B excludes A, A excludes B

Fallacy of Hasty Generalization EXAMPLE

To say that someone commits the fallacy of hasty generalization means that the person's estimate of the prevalence of some trait in a whole population, based on a small sample, is WRONG. False (good chance that COULD be wrong doesn't say that its wrong)

Independence

Two events are independent if NIETHER EVENT OCCURANCE CHANGES THE PROBABILITY OF THE OTHER EVENT. -So if A and B are independent then P(A/B)=P(A) and P(B/A)=P(B), which is to say that NIETHER EVENT AFFECT THE PROBABILITY OF THE OTHER -A independent of B -B independent of A

Interpretation Error

error in deciding what a statistic means

Negatation Rule

fundamental counting rule- if youre going to conduct three experiments, and if (A) things can happen in the 1st, (B) in the 2nd, (C) in the 3rd, then the total # of possible outcomes is (ABC)

Confidence level

how confident are you that the real population figure is within your confidence interval?

Dispersion of values

how spread out the values are

Fallacy of Hasty Generalization- How big should your sample be?

no single right answer depends on 1. population size 2.variability- more of the population variability= need larger sample (measured by confidence interval and margin of error) 3. How precise do you want your statistics to be (more precise- bigger sample) 4. How high do you want your confidence level to be? (More confidence- bigger sample) Note: first two not under your control last two up to you

Joint Exhaustive EXAMPLE

p. 191 6. If, in a fair lottery, one hundred tickets are sold and only one ticket is a winner, then if Jack and Jill each buy tickets the outcome that Jack wins and the outcome that Jill wins are- assuming they did not share any tickets-- but we cannot determine whether they are (?) because (?) whether they bought all of the tickets -There are fifty-two cards in a standard deck, and so on any single draw these fifty-two cards exhaust the possible outcomes of the draw, just as-- for all practical purposes-- Head and Tails exhaust the possible outcomes in a coin toss.

Independence EXAMPLE

p. 191 1. if i randomly draw a single card from the deck, then the outcome (jack) and the outcome of (diamond) are 3. If i make two random draws from a standard deck of cards with replacment of the first card, then (3 of hearts on the first draw) and (king of spades on the second draw are INDEPENDENT 7. A. successive draws from a deck of cards where each card is replaced afterwards and the deck is reshuffled- YES B. C. What you get when you flip a coin one hundred times and what you get on flip #101- YES (each ime you flip you flip for the first time) D. One of your parents is a genius, and you are a genius- not totally independent/ considers how smart parents are /genetics

Mutually exclusive EXAMPLE

p. 191 2. if i randomly draw a single card from a standard deck, then the outcome (spade) and the outcome of (diamond) are mutually exclusive outcomes P(S)= 1/4 =/(NOT) P(S/D)=0 (they are not equal)

Gambler Fallacy EXAMPLE

p.200 Ex. tossing a coin several times and getting tails each time, the probability of getting a tails again is the exact same as when you first flipped it 1/2

Confidence Interval

real population figure is in here falls in between variables

Mean

sum of values in a list divided by the number of values. 4,8,11,22,22 the mean is 67/5=13.4

Regression Fallacy

the fallacy of ignoring regression to the mean when we infer or explain why a statistical outlier was followed by a value closer to the average

General Conjunction Rule

the general formula for computing the probability has two events A and B will both occur at the same time

Odds example

the outcome of getting a 6 on one roll of a fair die =probability of the outcome is 1/6 -The odds against getting a 6 on one roll of a fair die are 5:1 ("five to one") since there are 5 unfavorable outcomes that are possible (5 outcome that do not involve getting a 6) and one favorable outcome -draw a card from a standard deck, then the odds against getting a jack are 48:4=12:1, since there are 48 none Jacks in a deck, and four jacks -When tossing a fair coin, the odds against getting heads or tails are even 1:1 -odds of getting a black card is also even, since they are 26:26= 1:1

Joint probability

the probability that BOTH A and B happen -P(A&B)

Selection effect

what happens when you draw a wrong conclusion about a population that results from the WAY you COLLECTED the sample (usually means you didnt collect it randomly) -when we reason from a sample that was not gathered randomly EXAMPLES 1,2,3,4,8 on PAGE 218