Untitled Flashcards Set

• Notation

  • Actual values of the response variable: y

  • Predicted value of the response variable: y-hat; ŷ

• Residual

  • Positive is point is above the line

  • Negative if point is below the line

  • e=y-y-hat

  • What would you get if you added up all the residuals from the scatterplot

    • Zero

• Example

  • 220lbs

  • -20lbs

  • Y, e

• How do we choose where the regression line goes

  • Regression line minimizes squared residuals

  • Least Squares Regression Line (LSRL) or line of best fit

• Line of best fit formula

  • y=bo+b1x

• Slope Formula

  • b1=rsysx

• How well does the regression line fit the data?

  • R2

    • Values are between 0 and +1

    • Represents the fraction of the variation(specifically the variance) in the response variable that is explained by the regression line

      • R2 close to 1 indicated the model explains a lot

    • R2=(correlation coefficient[r])2

• Practice

  • Predicted value

  • 40 units are explained

  • R2=40/50=0.80

  • r=(0.80)1/2=0.89

• Assumptions for regression

  • Quantitative variable condition

  • Straight enough condition

  • No outliers condition

  • “Does the Plot Thicken”

    • Residuals must have similar spread

    • Most common violation when residuals get more spread out

    • (P. 187)

    • Can check using a residual plot, plotting the residuals on the y-axis and the explanatory variable on the x-axis

• homoscedasticity/heteroskedasticity

• Regression models are appropriate only when they capture and underlying relationship

  •  Nothing interesting would be left behind

  • Residuals incorporate everything that is left behind

  • This means that the residuals should not be interesting

  • Plotting the residuals against the explanatory variable should show no relationship

  • (from p. 181)

• Standard Error:

  • Summarizes typical residual size

  • Rough estimate of how much the model is “off” by

• R2 revisited

  • R2 tells us the proportion of variation on the response variable that is explained by the explanatory variable

    • “Signal”

  • The leftover unexplained variation is summarized by the residuals

    • “Noise”

  • Total variance of the response variable = variance coming from the predicted response variable (from the regression model) + variance coming from the residuals

• Regression to the mean: when a sample is extreme, the next sample is likely to be closer to the mean

• “I trust Spike more than me”

• Joe Walch, 2024

• R2: The percentage of the variation in the response variable that is explained by the explanatory variable

• Total Variance= Unexplained Variance + Explained Variance

• To test whether the conditions for a regression are met, use a residual plot

  • Should see no patterns on the residual plot

• Shifting, rescaling and standardizing variables will not change correlation coefficient , but it will change slope and intercept

• Outliers, leverage and Influence

  • Outliers:

    • Large residuals

    • High leverage

  • Leverage:

    • Data points that are far from the mean

    • Will pull the line closer to themselves, making the residual deceptively small

  • Influential Point

    • If omitting a data point results in a model with a very different slope, than the point is influential

• Lurking variables can lead to spurious associations

• Regressions and causations

  • Regressions do not show causation

  • Be careful about lurking variables

  • Be careful when interpreting slopes

INTRO TO PROBABILITY

• Random Phenomena:

  • Situation where we know which outcomes could happen, but do not know which particular outcomes would happen

  • E.G. Coin Flip, drawing cards

• Trial:

  • A single attempt of a random phenomenon

  • E.G. A single coin flip

• Outcome:

  • Value that is measured, observed, or reported for a trial

• Event:

  • A collection of outcomes

  • Denoted with bold capital letters

  • E.G. flipping 2 coins and recording the outcomes

    • Getting a heads and heads in one event

• Sample Space:

  • Collection of all possible outcomes

  • Denoted with S={...}

  • E.G. flipping 2 coins

    • S = {HH, HT, TH, TT}

• What is the sample space for flipping 3 coins?

  • S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

• Law of large numbers:

  • The long-run relative frequency of repeated independent events get closer and closer to the true relative frequency as the number of trials increases

  • LLN

  • Sometimes mistakenly referred to as the “Law of Averages” which doesn’t exist

    • Gambler’s Fallacy

  • LLN only works over that long run; doesn’t say anything about the short run

    • “The house always wins”

• Probability:

  • Long run relative frequency if an event’s occurrence

    • Represented by a number between 1 and 0

    • Typically in decimal or fraction form

  • To denote the probability of event a occurring P(A)

    • If P(A)=1, than A will occur

    • If P(A)=0, than A will never occur

    • If P(A)=0.5, than A will occur half of the time over the long run

• Independance:

  • Two events are independent if learning that one event occurs does not change the probability of the other event occurring

• A fan might say that they are 40% sure that their team will win the game. Is that the same type of probability that we have been discussing

  • Subjective probability vs. Theoretical probability

    • Theoretical:

      • When a probability is based on a mathematical model

        • Fair coin toss/dice roll, shuffled deck of cards

    • Subjective:

      • Probability that represents someone’s personal degree of belief

        • “I’m 90% sure we will win the game”

• TREE DIAGRAM:

• 5 probability rules

  • Probability must be between 0 and 1

  • Probability Assignment rule

    • All probabilities must add up to 1

  • Complement rule

    • P(AC)=1-P(A)

    • Complement:

      • everything that is not in A is the complement of A

  • Addition rule

    • For two disjoint events A and B, the probability of that one or the other occurs is the sum of the two probabilities

• 5. 15.38461538%

• 6. Addition Rule(?)

• N.E.I.

• Addition Rule:

  • For two disjoint events A and B, the probability that one or the other occurs

•  Disjoint Events:

  • Events that have no outcomes in common

• General Addition Rule:

  • More flexible than addition rule

  • Used when events are not disjointed

  • Formal equation:

    • P(A ∪ B) = P(A) + P(B) – P(A ⋂ B)

• Conditional Probability:

  • The probability of an event given the occurrence of another event

  • Probability applied to a conditional distribution

  • P(B | A)=P(A∩B)/P(A)

    • Probability of B, conditioned on A

      • A “given” B

  • independent when

    • P(B | A)=P(B)

      • ஃ A & B are independent

• Venn Diagram

  • Uses both a rectangle and some circles

• General Product Rule

  • P(A⋂B)=P(A) * P(B | A)

• →Disjoint/independent events are required to use simple addition/multiplication rule

• Random Variable:

  • Variable whose value depends on a random event

  • Denoted by ‘X’

  • Values are denoted by ‘x’

  • E.G. coin flips, dice rolls, card draws, etc.

• Probability Model:

  • Function that associates a probability with each value of a discrete random variable

  • Typically in a table form with at least 3 columns

• Expected Value:

  • Theoretical long run average of a random variable

  • Center of a probability model for the random variable(like the mean)

  • Denoted by E(X) or μ

  • Calculated by the sum of the products of variable values and probabilities

  • Analogous to the ”break even” point or house edge

• Random:

  • An outcome is random if we know the possible outcomes but not which value it actually takes

  • Random outcomes are free of human influence

  • Don’t use “random” in place of “unexpected”

  • Examples

    • “Random” phone call

    • “Random” actions

• Simulation: Using random numbers to represent the outcomes of uncertain events

• Trial:

  • In a simulation, the sequence of events that we are pretending will take place

  • For each trial, we get a simulated answer to our question(simulated outcome)

• DISCRETE VS CONtINUOUS

  • D - finite number

  • C - any within interval

• Bernoulli Trials:

  • Collection of trials where trial:

    • Each has exactly two outcomes: “success” or “failure”

      • q: success

      • p: failure

    • P(“success”) is constant

    • All trials are independent

• Geometric Probability Model:

  • Used with random variables that count the number of Bernoulli trials until our first success

  • X = the number of trials until the first success

  • p = the probability of success

  • q = the probability of failure

    • q=1-p

  • p and q are compliments

  • P(X=x)=qx-1p

  • E(X)==1p

    • Note→on the ap exam, 1-p will be shown instead of q

  • Var(X)=qp2

  • Standard Deviation ==qp2

• 10% Condition

  • Remember that one of the requirements for Bernouli trials is independence, and trials are not independent when we sample without replacement

  • However, it is still ok to use this model as long as we randomly sample less than 10% of the population

• Binomial Model:

  • Appropriate for a random variable that counts the number of successes in a fixed number of Bernoulli Trials

  • Example: getting 2 heads with 4 coin flips

  • Probability of getting x successes in n trials

  • Details:

    • x → number of successes

    • n → number of trials

    • p → probability of success (1-q)

    • q → probability of failure (1-p)

    • P(x)=n!x!(n-x)!pxqn-x

    • Var(X)=npq

    • SD(X)=npq

• Systematic Sample

  • SImple Random Sample, SMS is the gold standard, but not often the most practical

    • Systematic Sample - 

      • Still has randomness, but each is not equally likely

    • Stratified Random Sample

      • Population divided into several subpopulations

      • SRS within each strata

      • Used by differences in the subgroups and want to capture those differences proportionally

    • Cluster Sample

      • Population is divided into groups or clusters

      • Each cluster is similar to other clusters

      • Done for convenience, practicality and/or cost

    • Multistage Sampling

      • Combo of multiple methods (usually Stratified and Cluster)

      • EG

        • For Kauai, we can stratify by moku, than cluster by neighborhood or city block

• Surveys:

  • How are you asking your questions?

  • Specific questions

  • Careful with phrasing

  • See p. 290-291

  • Pilot Survey:

    • Small Trial run of a Survey to test whether the questions and setup are good and clear

• What can go wrong?

  • Voluntary response sample:

    • A large group is invited to respond and anyone who chooses to respond are counted

    • Leads to a Voluntary Response Bias:

      • Example: Very strongly opinionated people might be more likely to volunteer

  • Convenience sample:

    • D

  • Bad Sampling Coverage:

    • If the sampling frame excludes people from the population

  • Undercoverage:

    • Minorities during the census

  • Nonresponse Bias: bias introduced when a large fraction of those sampled fails to respond to a survey

  • Response Bias: Anything in a survey that influences responses (like leading questions or unclear phrasing)

• The Success/Failure Condition:

  • A binomial model is approximately normal if we expect at least 10 successes and 10 failures

    • np10

    • nq10

• Discrete vs Continuous models

  • Normal Distribution is continuous

  • Binomial model is discrete

• Statistical Significance: The results of a study are considered statistically significant is there is a very low probability that they happened by chance

  • Are the results extreme enough to reject a hypothesis?

• Sampling Distribution

  • Distribution of sample means

• Complement Rule:

  • P(AC)=1-P(A)

  • Complement:

    • everything that is not in A is the complement of A

• Addition Rule:

  • For two disjoint events A and B, the probability of that one or the other occurs is the sum of the two probabilities

•  Disjoint Events:

  • Events that have no outcomes in common

• General Addition Rule:

  • More flexible than addition rule

  • Used when events are not disjointed

  • Formal equation:

    • P(A ∪ B) = P(A) + P(B) – P(A ⋂ B)

• Conditional Probability:

  • The probability of an event given the occurrence of another event

  • Probability applied to a conditional distribution

  • P(B | A)=P(A∩B)/P(A)

    • Probability of B, conditioned on A

      • A “given” B

  • independent when

    • P(B | A)=P(B)

      • ஃ A & B are independent

• Shifting data affects center but not spread

  • E(X+C)=E(X)+C

  • E(X±Y)=E(X)±E(Y)

  • SD(X+C)=SD(X) (same standard deviation)

  • Var(X+C)=Var(X) (same variance)

  • Var(X±Y)=Var(X)+Var(Y)

• Rescaling data affects center and spread

  • E(X*C)=E(X)*C

  • SD(X*C)=SD(X)*C

  • Var(X*C)=Var(X)*C^2This relationship demonstrates how variance scales with the square of the constant factor, indicating that as we multiply a random variable by a constant, the variability increases in proportion to the square of that constant.