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This document provides an overview of key concepts in statistics and probability, including: - Population and sample definitions and sampling techniques - Descriptive statistics such as mean, median, variance, and standard deviation - Probability rules and concepts like independent and mutually exclusive events - Common probability distributions like the binomial, normal, and uniform distributions - Hypothesis testing methods using confidence intervals and p-values - Simple linear regression to model relationships between variables The document covers a wide range of foundational statistical topics in a detailed yet accessible manner.

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Population ­ entire collection of objects or individuals about which information is desired. ➔ easier to take a sample ◆ Sample ­ part of the population that is selected for analysis ◆ Watch out for: ● Limited sample size that might not be representative of population ◆ Simple Random Sampling­ Every possible sample of a certain size has the same chance of being selected Observational Study ­ there can always be lurking variables affecting results ➔ i.e, strong positive association between shoe size and intelligence for boys ➔ **should never show causation Experimental Study­ lurking variables can be controlled; can give good evidence for causation Descriptive Statistics Part I ➔ Summary Measures ➔ Mean ­ arithmetic average of data values ◆ **Highly susceptible to extreme values (outliers). Goes towards extreme values ◆ Mean could never be larger or smaller than max/min value but could be the max/min value ➔ Median ­ in an ordered array, the median is the middle number ◆ **Not affected by extreme values ➔ Quartiles ­ split the ranked data into 4 equal groups ◆ Box and Whisker Plot ➔ Range = Xmaximum Xminimum ◆ Disadvantages: Ignores the way in which data are distributed; sensitive to outliers ➔ Interquartile Range (IQR) = 3rd quartile ­ 1st quartile ◆ Not used that much ◆ Not affected by outliers ➔ Variance ­ the average distance squared sx 2 = n 1 (x x) ∑ n i=1 i 2 ◆ gets rid of the negative sx 2 values ◆ units are squared ➔ Standard Deviation ­ shows variation about the mean s = √ n 1 (x x) ∑ n i=1 i 2 ◆ highly affected by outliers ◆ has same units as original data ◆ finance = horrible measure of risk (trampoline example) Descriptive Statistics Part II Linear Transformations ➔ Linear transformations change the center and spread of data ➔ ar(a X) V ar(X) V + b = b2 ➔ Average(a+bX) = a+b[Average(X)] STAT 100 Final Cheat Sheets - Harvard University
➔ Effects of Linear Transformations: ◆ a + b*mean meannew = ◆ a + b*median mediannew = ◆ *stdev stdevnew = b | | ◆ *IQR IQRnew = b | | ➔ Z­score ­ new data set will have mean 0 and variance 1 z = S X X Empirical Rule ➔ Only for mound­shaped data Approx. 95% of data is in the interval: x s , x s ) s ( 2 x + 2 x = x + / 2 x ➔ only use if you just have mean and std. dev. Chebyshev's Rule ➔ Use for any set of data and for any number k, greater than 1 (1.2, 1.3, etc.) ➔ 1 1 k2 ➔ (Ex) for k=2 (2 standard deviations), 75% of data falls within 2 standard deviations Detecting Outliers ➔ Classic Outlier Detection ◆ doesn't always work ◆ z | | = | | S X X | | ≥ 2 ➔ The Boxplot Rule ◆ Value X is an outlier if: X<Q1­1.5(Q3­Q1) or X>Q3+1.5(Q3­Q1) Skewness ➔ measures the degree of asymmetry exhibited by data ◆ negative values= skewed left ◆ positive values= skewed right ◆ if = don't need .8 skewness | | < 0 to transform data Measurements of Association ➔ Covariance ◆ Covariance > 0 = larger x, larger y ◆ Covariance < 0 = larger x, smaller y ◆ s (x )(y ) xy = 1 n 1 ∑ n i=1 x y ◆ Units = Units of x Units of y ◆ Covariance is only +, ­, or 0 (can be any number) ➔ Correlation ­ measures strength of a linear relationship between two variables ◆ rxy = covariancexy (std.dev. )(std. dev. ) x y ◆ correlation is between ­1 and 1 ◆ Sign: direction of relationship ◆ Absolute value: strength of relationship (­0.6 is stronger relationship than +0.4) ◆ Correlation doesn't imply causation ◆ The correlation of a variable with itself is one Combining Data Sets ➔ Mean (Z) = X Y Z = a + b ➔ Var (Z) = V ar(X) V ar(Y ) sz 2 = a2 + b2 + abCov(X, ) 2 Y Portfolios ➔ Return on a portfolio: R R Rp = wA A + wB B ◆ weights add up to 1 ◆ return = mean ◆ risk = std. deviation ➔ Variance of return of portfolio s s s2 p = w2 A 2 A + w2 B 2 B + w w (s ) 2 A B A,B ◆ Risk(variance) is reduced when stocks are negatively correlated. (when there's a negative covariance) Probability ➔ measure of uncertainty ➔ all outcomes have to be exhaustive (all options possible) and mutually exhaustive (no 2 outcomes can occur at the same time)
Probability Rules 1. Probabilities range from rob(A) 0 ≤ P ≤ 1 2. The probabilities of all outcomes must add up to 1 3. The complement rule = A happens or A doesn't happen (A) (A) P = 1 P (A) (A) P + P = 1 4. Addition Rule: (A or B) (A) (B) (A and B) P = P + P P Contingency/Joint Table ➔ To go from contingency to joint table, divide by total # of counts ➔ everything inside table adds up to 1 Conditional Probability ➔ (A|B) P ➔ (A|B) P = P(B) P(A and B) ➔ Given event B has happened, what is the probability event A will happen? ➔ Look out for: "given", "if" Independence ➔ Independent if: or (A|B) (A) P = P (B|A) (B) P = P ➔ If probabilities change, then A and B are dependent ➔ **hard to prove independence, need to check every value Multiplication Rules ➔ If A and B are INDEPENDENT: (A and B) (A) (B) P = P P ➔ Another way to find joint probability: (A and B) (A|B) (B) P = P P (A and B) (B|A) (A) P = P P 2 x 2 Table Decision Analysis ➔ Maximax solution = optimistic approach. Always think the best is going to happen ➔ Maximin solution = pessimistic approach. ➔ Expected Value Solution = MV (P ) (P )... (P ) E = X1 1 + X2 2 + Xn n Decision Tree Analysis ➔ square = your choice ➔ circle = uncertain events Discrete Random Variables ➔ (x) (X ) PX = P = x Expectation ➔ = E(x) = μx P(X ) ∑xi = xi ➔ Example: 2)(0.1) 3)(0.5) .7 ( + ( = 1 Variance ➔ (x ) σ2 = E 2 μx 2 ➔ Example: 2) (0.1) 3) (0.5) 1.7) .01 ( 2 + ( 2 ( 2 = 2 Rules for Expectation and Variance ➔ (s) a bμ μs = E = + x ➔ Var(s)= b2 σ2 Jointly Distributed Discrete Random Variables ➔ Independent if: (X and Y ) (x) (y) Px,y = x = y = Px Py
➔ Combining Random Variables ◆ If X and Y are independent: (X ) (X) (Y ) E + Y = E + E ar(X ) ar(X) ar(Y ) V + Y = V + V ◆ If X and Y are dependent: (X ) (X) (Y ) E + Y = E + E ar(X ) ar(X) ar(Y ) Cov(X, ) V + Y = V + V + 2 Y ➔ Covariance: ov(X, ) (XY ) (X)E(Y ) C Y = E E ➔ If X and Y are independent, Cov(X,Y) = 0 Binomial Distribution ➔ doing something n times ➔ only 2 outcomes: success or failure ➔ trials are independent of each other ➔ probability remains constant 1.) All Failures (all failures) 1 ) P = ( p n 2.) All Successes (all successes) P = pn 3.) At least one success (at least 1 success) 1 ) P = 1 ( p n 4.) At least one failure P(at least 1 failure) = 1 pn 5.) Binomial Distribution Formula for x=exact value 6.) Mean (Expectation) (x) p μ = E = n 7.) Variance and Standard Dev. pq σ2 = n σ = √npq q = 1 p Binomial Example Continuous Probability Distributions ➔ the probability that a continuous random variable X will assume any particular value is 0 ➔ Density Curves ◆ Area under the curve is the probability that any range of values will occur. ◆ Total area = 1 Uniform Distribution ◆ nif (a, ) X ~ U b Uniform Example (Example cont'd next page)
➔ Mean for uniform distribution: (X) E = 2 (a+b) ➔ Variance for unif. distribution: ar(X) V = 12 (b a)2 Normal Distribution ➔ governed by 2 parameters: (the mean) and (the standard μ σ deviation) ➔ (μ, ) X ~ N σ2 Standardize Normal Distribution: Z = σ X μ ➔ Z­score is the number of standard deviations the related X is from its mean ➔ **Z< some value, will just be the probability found on table ➔ **Z> some value, will be (1­probability) found on table Normal Distribution Example Sums of Normals Sums of Normals Example: ➔ Cov(X,Y) = 0 b/c they're independent Central Limit Theorem ➔ as n increases, ➔ should get closer to (population x μ mean) ➔ mean( ) x = μ ➔ variance x) n ( = σ2 / ➔ (μ, ) X ~ N n σ2 ◆ if population is normally distributed, n can be any value ◆ any population, n needs to be 0 ≥ 3 ➔ Z = X μ σ/√n Confidence Intervals = tells us how good our estimate is **Want high confidence, narrow interval **As confidence increases , interval also increases A. One Sample Proportion ➔ p ︿ = x n = sample size number of successes in sample ➔ ➔ We are thus 95% confident that the true population proportion is in the interval… ➔ We are assuming that n is large, n >5 and p ︿ our sample size is less than 10% of the population size.
Standard Error and Margin of Error Example of Sample Proportion Problem Determining Sample Size n = e2 (1.96) p(1 p) 2︿ ︿ ➔ If given a confidence interval, is p ︿ the middle number of the interval ➔ No confidence interval; use worst case scenario ◆ =0.5 p ︿ B. One Sample Mean For samples n > 30 Confidence Interval: ➔ If n > 30, we can substitute s for so that we get: σ For samples n < 30 T Distribution used when: ➔ is not known, n < 30, and data is σ normally distributed *Stata always uses the t­distribution when computing confidence intervals Hypothesis Testing ➔ Null Hypothesis: ➔ , a statement of no change and is H0 assumed true until evidence indicates otherwise. ➔ Alternative Hypothesis: is a Ha statement that we are trying to find evidence to support. ➔ Type I error: reject the null hypothesis when the null hypothesis is true. (considered the worst error) ➔ Type II error: do not reject the null hypothesis when the alternative hypothesis is true. Example of Type I and Type II errors Methods of Hypothesis Testing 1. Confidence Intervals ** 2. Test statistic 3. P­values ** ➔ C.I and P­values always safe to do because don’t need to worry about size of n (can be bigger or smaller than 30)
One Sample Hypothesis Tests 1. Confidence Interval (can be used only for two­sided tests) 2. Test Statistic Approach (Population Mean) 3. Test Statistic Approach (Population Proportion) 4. P­Values ➔ a number between 0 and 1 ➔ the larger the p­value, the more consistent the data is with the null ➔ the smaller the p­value, the more consistent the data is with the alternative ➔ **If P is low (less than 0.05), must go ­ reject the null H0 hypothesis
Two Sample Hypothesis Tests 1. Comparing Two Proportions (Independent Groups) ➔ Calculate Confidence Interval ➔ Test Statistic for Two Proportions 2. Comparing Two Means (large independent samples n>30) ➔ Calculating Confidence Interval ➔ Test Statistic for Two Means Matched Pairs ➔ Two samples are DEPENDENT Example:
Simple Linear Regression ➔ used to predict the value of one variable (dependent variable) on the basis of other variables (independent variables) ➔ X Y ︿ = b0 + b1 ➔ Residual: e = Y Y ︿ fitted ➔ Fitting error: X ei = Y i Y i ︿ = Y i b0 bi i ◆ e is the part of Y not related to X ➔ Values of and which minimize b0 b1 the residual sum of squares are: (slope) b1 = rsx sy X b0 = Y b1 ➔ Interpretation of slope ­ for each additional x value (e.x. mile on odometer), the y value decreases/ increases by an average of value b1 ➔ Interpretation of y­intercept ­ plug in 0 for x and the value you get for is y ︿ the y­intercept (e.x. y=3.25­0.0614xSkippedClass, a student who skips no classes has a gpa of 3.25.) ➔ **danger of extrapolation ­ if an x value is outside of our data set, we can't confidently predict the fitted y value Properties of the Residuals and Fitted Values 1. Mean of the residuals = 0; Sum of the residuals = 0 2. Mean of original values is the same as mean of fitted values Y = Y ︿ 3. 4. Correlation Matrix ➔ corr Y , ) ( ︿ e = 0 A Measure of Fit: R2 ➔ Good fit: if SSR is big, SEE is small ➔ SST=SSR, perfect fit ➔ : coefficient of determination R2 R2 = SSR SST = 1 SST SSE ➔ is between 0 and 1, the closer R2 R2 is to 1, the better the fit ➔ Interpretation of : (e.x. 65% of the R2 variation in the selling price is explained by the variation in odometer reading. The rest 35% remains unexplained by this model) ➔ ** doesn’t indicate whether model R2 is adequate** ➔ As you add more X’s to model, R2 goes up ➔ Guide to finding SSR, SSE, SST
Assumptions of Simple Linear Regression 1. We model the AVERAGE of something rather than something itself 2. ◆ As (noise) gets bigger, it’s ε harder to find the line Estimating Se ➔ S2 e = n 2 SSE ➔ is our estimate of σ Se 2 2 ➔ is our estimate of σ Se = √Se 2 ➔ 95% of the Y values should lie within the interval X 1.96S b0 + b1 + e Example of Prediction Intervals: Standard Errors for and b b1 0 ➔ standard errors when noise ➔ amount of uncertainty in our sb0 estimate of (small s good, large s β0 bad) ➔ amount of uncertainty in our sb1 estimate of β1 Confidence Intervals for and b b1 0 ➔ ➔ ➔ ➔ ➔ n small → bad big → bad se small→ bad (wants x’s spread out for s2 x better guess) Regression Hypothesis Testing *always a two­sided test ➔ want to test whether slope ( ) is β1 needed in our model ➔ : = 0 (don’t need x) H0 β1 : 0 (need x) Ha = β1 / ➔ Need X in the model if: a. 0 isn’t in the confidence interval b. t > 1.96 c. P­value < 0.05 Test Statistic for Slope/Y­intercept ➔ can only be used if n>30 ➔ if n < 30, use p­values
Multiple Regression ➔ ➔ Variable Importance: ◆ higher t­value, lower p­value = variable is more important ◆ lower t­value, higher p­value = variable is less important (or not needed) Adjusted R­squared ➔ k = # of X’s ➔ Adj. R­squared will as you add junk x variables ➔ Adj. R­squared will only if the x you add in is very useful ➔ **want Adj. R­squared to go up and Se low for better model The Overall F Test ➔ Always want to reject F test (reject null hypothesis) ➔ Look at p­value (if < 0.05, reject null) ➔ : (don’t H0 ... β1 = β2 = β3 = βk = 0 need any X’s) : (need at Ha ... = β1 = β2 = β3 = βk / 0 least 1 X) ➔ If no x variables needed, then SSR=0 and SST=SSE Modeling Regression Backward Stepwise Regression 1. Start will all variables in the model 2. at each step, delete the least important variable based on largest p­value above 0.05 3. stop when you can’t delete anymore ➔ Will see Adj. R­squared and Se Dummy Variables ➔ An indicator variable that takes on a value of 0 or 1, allow intercepts to change Interaction Terms ➔ allow the slopes to change ➔ interaction between 2 or more x variables that will affect the Y variable How to Create Dummy Variables (Nominal Variables) ➔ If C is the number of categories, create (C­1) dummy variables for describing the variable ➔ One category is always the “baseline”, which is included in the intercept Recoding Dummy Variables Example: How many hockey sticks sold in the summer (original equation) ockey 00 0Wtr 0Spr 0Fall h = 1 + 1 2 + 3 Write equation for how many hockey sticks sold in the winter ockey 10 0Fall 0Spri 0Summer h = 1 + 2 3 1 ➔ **always need to get same exact values from the original equation
Regression Diagnostics Standardize Residuals Check Model Assumptions ➔ Plot residuals versus Yhat ➔ Outliers ◆ Regression likes to move towards outliers (shows up as being really high) R2 ◆ want to remove outlier that is extreme in both x and y ➔ Nonlinearity (ovtest) ◆ Plotting residuals vs. fitted values will show a relationship if data is nonlinear ( also high) R2 ◆ Log transformation ­ accommodates non­linearity, reduces right skewness in the Y, eliminates heteroskedasticity ◆ **Only take log of X variable so that we can compare models. Can’t compare models if you take log of Y. ◆ Transformations cheatsheet ◆ ovtest: a significant test statistic indicates that polynomial terms should be added ◆ : H0 ata o transformation d = n : Ha ata = o transformation d / n ➔ Normality (sktest) ◆ : H0 ata ormality d = n : Ha ata = ormality d / n ◆ don’t want to reject the null hypothesis. P­value should be big ➔ Homoskedasticity (hettest) ◆ : H0 ata omoskedasticity d = h ◆ ata = omoskedasticity Ha : d / h ◆ Homoskedastic: band around the values ◆ Heteroskedastic: as x goes up, the noise goes up (no more band, fan­shaped) ◆ If heteroskedastic, fix it by logging the Y variable ◆ If heteroskedastic, fix it by making standard errors robust ➔ Multicollinearity ◆ when x variables are highly correlated with each other. ◆ > 0.9 R2 ◆ pairwise correlation > 0.9 ◆ correlate all x variables, include y variable, drop the x variable that is less correlated to y Summary of Regression Output Source: http://people.fas.harvard.edu/~mparzen/stat100/Stat%20100%20Final%20Cheat%20Sheets%20Docs%20(2).pdf

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Statistics_summary_1634533932.pdf

  • 1.
    Population ­ entirecollection of objects or individuals about which information is desired. ➔ easier to take a sample ◆ Sample ­ part of the population that is selected for analysis ◆ Watch out for: ● Limited sample size that might not be representative of population ◆ Simple Random Sampling­ Every possible sample of a certain size has the same chance of being selected Observational Study ­ there can always be lurking variables affecting results ➔ i.e, strong positive association between shoe size and intelligence for boys ➔ **should never show causation Experimental Study­ lurking variables can be controlled; can give good evidence for causation Descriptive Statistics Part I ➔ Summary Measures ➔ Mean ­ arithmetic average of data values ◆ **Highly susceptible to extreme values (outliers). Goes towards extreme values ◆ Mean could never be larger or smaller than max/min value but could be the max/min value ➔ Median ­ in an ordered array, the median is the middle number ◆ **Not affected by extreme values ➔ Quartiles ­ split the ranked data into 4 equal groups ◆ Box and Whisker Plot ➔ Range = Xmaximum Xminimum ◆ Disadvantages: Ignores the way in which data are distributed; sensitive to outliers ➔ Interquartile Range (IQR) = 3rd quartile ­ 1st quartile ◆ Not used that much ◆ Not affected by outliers ➔ Variance ­ the average distance squared sx 2 = n 1 (x x) ∑ n i=1 i 2 ◆ gets rid of the negative sx 2 values ◆ units are squared ➔ Standard Deviation ­ shows variation about the mean s = √ n 1 (x x) ∑ n i=1 i 2 ◆ highly affected by outliers ◆ has same units as original data ◆ finance = horrible measure of risk (trampoline example) Descriptive Statistics Part II Linear Transformations ➔ Linear transformations change the center and spread of data ➔ ar(a X) V ar(X) V + b = b2 ➔ Average(a+bX) = a+b[Average(X)] STAT 100 Final Cheat Sheets - Harvard University
  • 2.
    ➔ Effects ofLinear Transformations: ◆ a + b*mean meannew = ◆ a + b*median mediannew = ◆ *stdev stdevnew = b | | ◆ *IQR IQRnew = b | | ➔ Z­score ­ new data set will have mean 0 and variance 1 z = S X X Empirical Rule ➔ Only for mound­shaped data Approx. 95% of data is in the interval: x s , x s ) s ( 2 x + 2 x = x + / 2 x ➔ only use if you just have mean and std. dev. Chebyshev's Rule ➔ Use for any set of data and for any number k, greater than 1 (1.2, 1.3, etc.) ➔ 1 1 k2 ➔ (Ex) for k=2 (2 standard deviations), 75% of data falls within 2 standard deviations Detecting Outliers ➔ Classic Outlier Detection ◆ doesn't always work ◆ z | | = | | S X X | | ≥ 2 ➔ The Boxplot Rule ◆ Value X is an outlier if: X<Q1­1.5(Q3­Q1) or X>Q3+1.5(Q3­Q1) Skewness ➔ measures the degree of asymmetry exhibited by data ◆ negative values= skewed left ◆ positive values= skewed right ◆ if = don't need .8 skewness | | < 0 to transform data Measurements of Association ➔ Covariance ◆ Covariance > 0 = larger x, larger y ◆ Covariance < 0 = larger x, smaller y ◆ s (x )(y ) xy = 1 n 1 ∑ n i=1 x y ◆ Units = Units of x Units of y ◆ Covariance is only +, ­, or 0 (can be any number) ➔ Correlation ­ measures strength of a linear relationship between two variables ◆ rxy = covariancexy (std.dev. )(std. dev. ) x y ◆ correlation is between ­1 and 1 ◆ Sign: direction of relationship ◆ Absolute value: strength of relationship (­0.6 is stronger relationship than +0.4) ◆ Correlation doesn't imply causation ◆ The correlation of a variable with itself is one Combining Data Sets ➔ Mean (Z) = X Y Z = a + b ➔ Var (Z) = V ar(X) V ar(Y ) sz 2 = a2 + b2 + abCov(X, ) 2 Y Portfolios ➔ Return on a portfolio: R R Rp = wA A + wB B ◆ weights add up to 1 ◆ return = mean ◆ risk = std. deviation ➔ Variance of return of portfolio s s s2 p = w2 A 2 A + w2 B 2 B + w w (s ) 2 A B A,B ◆ Risk(variance) is reduced when stocks are negatively correlated. (when there's a negative covariance) Probability ➔ measure of uncertainty ➔ all outcomes have to be exhaustive (all options possible) and mutually exhaustive (no 2 outcomes can occur at the same time)
  • 3.
    Probability Rules 1. Probabilitiesrange from rob(A) 0 ≤ P ≤ 1 2. The probabilities of all outcomes must add up to 1 3. The complement rule = A happens or A doesn't happen (A) (A) P = 1 P (A) (A) P + P = 1 4. Addition Rule: (A or B) (A) (B) (A and B) P = P + P P Contingency/Joint Table ➔ To go from contingency to joint table, divide by total # of counts ➔ everything inside table adds up to 1 Conditional Probability ➔ (A|B) P ➔ (A|B) P = P(B) P(A and B) ➔ Given event B has happened, what is the probability event A will happen? ➔ Look out for: "given", "if" Independence ➔ Independent if: or (A|B) (A) P = P (B|A) (B) P = P ➔ If probabilities change, then A and B are dependent ➔ **hard to prove independence, need to check every value Multiplication Rules ➔ If A and B are INDEPENDENT: (A and B) (A) (B) P = P P ➔ Another way to find joint probability: (A and B) (A|B) (B) P = P P (A and B) (B|A) (A) P = P P 2 x 2 Table Decision Analysis ➔ Maximax solution = optimistic approach. Always think the best is going to happen ➔ Maximin solution = pessimistic approach. ➔ Expected Value Solution = MV (P ) (P )... (P ) E = X1 1 + X2 2 + Xn n Decision Tree Analysis ➔ square = your choice ➔ circle = uncertain events Discrete Random Variables ➔ (x) (X ) PX = P = x Expectation ➔ = E(x) = μx P(X ) ∑xi = xi ➔ Example: 2)(0.1) 3)(0.5) .7 ( + ( = 1 Variance ➔ (x ) σ2 = E 2 μx 2 ➔ Example: 2) (0.1) 3) (0.5) 1.7) .01 ( 2 + ( 2 ( 2 = 2 Rules for Expectation and Variance ➔ (s) a bμ μs = E = + x ➔ Var(s)= b2 σ2 Jointly Distributed Discrete Random Variables ➔ Independent if: (X and Y ) (x) (y) Px,y = x = y = Px Py
  • 4.
    ➔ Combining RandomVariables ◆ If X and Y are independent: (X ) (X) (Y ) E + Y = E + E ar(X ) ar(X) ar(Y ) V + Y = V + V ◆ If X and Y are dependent: (X ) (X) (Y ) E + Y = E + E ar(X ) ar(X) ar(Y ) Cov(X, ) V + Y = V + V + 2 Y ➔ Covariance: ov(X, ) (XY ) (X)E(Y ) C Y = E E ➔ If X and Y are independent, Cov(X,Y) = 0 Binomial Distribution ➔ doing something n times ➔ only 2 outcomes: success or failure ➔ trials are independent of each other ➔ probability remains constant 1.) All Failures (all failures) 1 ) P = ( p n 2.) All Successes (all successes) P = pn 3.) At least one success (at least 1 success) 1 ) P = 1 ( p n 4.) At least one failure P(at least 1 failure) = 1 pn 5.) Binomial Distribution Formula for x=exact value 6.) Mean (Expectation) (x) p μ = E = n 7.) Variance and Standard Dev. pq σ2 = n σ = √npq q = 1 p Binomial Example Continuous Probability Distributions ➔ the probability that a continuous random variable X will assume any particular value is 0 ➔ Density Curves ◆ Area under the curve is the probability that any range of values will occur. ◆ Total area = 1 Uniform Distribution ◆ nif (a, ) X ~ U b Uniform Example (Example cont'd next page)
  • 5.
    ➔ Mean foruniform distribution: (X) E = 2 (a+b) ➔ Variance for unif. distribution: ar(X) V = 12 (b a)2 Normal Distribution ➔ governed by 2 parameters: (the mean) and (the standard μ σ deviation) ➔ (μ, ) X ~ N σ2 Standardize Normal Distribution: Z = σ X μ ➔ Z­score is the number of standard deviations the related X is from its mean ➔ **Z< some value, will just be the probability found on table ➔ **Z> some value, will be (1­probability) found on table Normal Distribution Example Sums of Normals Sums of Normals Example: ➔ Cov(X,Y) = 0 b/c they're independent Central Limit Theorem ➔ as n increases, ➔ should get closer to (population x μ mean) ➔ mean( ) x = μ ➔ variance x) n ( = σ2 / ➔ (μ, ) X ~ N n σ2 ◆ if population is normally distributed, n can be any value ◆ any population, n needs to be 0 ≥ 3 ➔ Z = X μ σ/√n Confidence Intervals = tells us how good our estimate is **Want high confidence, narrow interval **As confidence increases , interval also increases A. One Sample Proportion ➔ p ︿ = x n = sample size number of successes in sample ➔ ➔ We are thus 95% confident that the true population proportion is in the interval… ➔ We are assuming that n is large, n >5 and p ︿ our sample size is less than 10% of the population size.
  • 6.
    Standard Error andMargin of Error Example of Sample Proportion Problem Determining Sample Size n = e2 (1.96) p(1 p) 2︿ ︿ ➔ If given a confidence interval, is p ︿ the middle number of the interval ➔ No confidence interval; use worst case scenario ◆ =0.5 p ︿ B. One Sample Mean For samples n > 30 Confidence Interval: ➔ If n > 30, we can substitute s for so that we get: σ For samples n < 30 T Distribution used when: ➔ is not known, n < 30, and data is σ normally distributed *Stata always uses the t­distribution when computing confidence intervals Hypothesis Testing ➔ Null Hypothesis: ➔ , a statement of no change and is H0 assumed true until evidence indicates otherwise. ➔ Alternative Hypothesis: is a Ha statement that we are trying to find evidence to support. ➔ Type I error: reject the null hypothesis when the null hypothesis is true. (considered the worst error) ➔ Type II error: do not reject the null hypothesis when the alternative hypothesis is true. Example of Type I and Type II errors Methods of Hypothesis Testing 1. Confidence Intervals ** 2. Test statistic 3. P­values ** ➔ C.I and P­values always safe to do because don’t need to worry about size of n (can be bigger or smaller than 30)
  • 7.
    One Sample HypothesisTests 1. Confidence Interval (can be used only for two­sided tests) 2. Test Statistic Approach (Population Mean) 3. Test Statistic Approach (Population Proportion) 4. P­Values ➔ a number between 0 and 1 ➔ the larger the p­value, the more consistent the data is with the null ➔ the smaller the p­value, the more consistent the data is with the alternative ➔ **If P is low (less than 0.05), must go ­ reject the null H0 hypothesis
  • 8.
    Two Sample HypothesisTests 1. Comparing Two Proportions (Independent Groups) ➔ Calculate Confidence Interval ➔ Test Statistic for Two Proportions 2. Comparing Two Means (large independent samples n>30) ➔ Calculating Confidence Interval ➔ Test Statistic for Two Means Matched Pairs ➔ Two samples are DEPENDENT Example:
  • 9.
    Simple Linear Regression ➔used to predict the value of one variable (dependent variable) on the basis of other variables (independent variables) ➔ X Y ︿ = b0 + b1 ➔ Residual: e = Y Y ︿ fitted ➔ Fitting error: X ei = Y i Y i ︿ = Y i b0 bi i ◆ e is the part of Y not related to X ➔ Values of and which minimize b0 b1 the residual sum of squares are: (slope) b1 = rsx sy X b0 = Y b1 ➔ Interpretation of slope ­ for each additional x value (e.x. mile on odometer), the y value decreases/ increases by an average of value b1 ➔ Interpretation of y­intercept ­ plug in 0 for x and the value you get for is y ︿ the y­intercept (e.x. y=3.25­0.0614xSkippedClass, a student who skips no classes has a gpa of 3.25.) ➔ **danger of extrapolation ­ if an x value is outside of our data set, we can't confidently predict the fitted y value Properties of the Residuals and Fitted Values 1. Mean of the residuals = 0; Sum of the residuals = 0 2. Mean of original values is the same as mean of fitted values Y = Y ︿ 3. 4. Correlation Matrix ➔ corr Y , ) ( ︿ e = 0 A Measure of Fit: R2 ➔ Good fit: if SSR is big, SEE is small ➔ SST=SSR, perfect fit ➔ : coefficient of determination R2 R2 = SSR SST = 1 SST SSE ➔ is between 0 and 1, the closer R2 R2 is to 1, the better the fit ➔ Interpretation of : (e.x. 65% of the R2 variation in the selling price is explained by the variation in odometer reading. The rest 35% remains unexplained by this model) ➔ ** doesn’t indicate whether model R2 is adequate** ➔ As you add more X’s to model, R2 goes up ➔ Guide to finding SSR, SSE, SST
  • 10.
    Assumptions of SimpleLinear Regression 1. We model the AVERAGE of something rather than something itself 2. ◆ As (noise) gets bigger, it’s ε harder to find the line Estimating Se ➔ S2 e = n 2 SSE ➔ is our estimate of σ Se 2 2 ➔ is our estimate of σ Se = √Se 2 ➔ 95% of the Y values should lie within the interval X 1.96S b0 + b1 + e Example of Prediction Intervals: Standard Errors for and b b1 0 ➔ standard errors when noise ➔ amount of uncertainty in our sb0 estimate of (small s good, large s β0 bad) ➔ amount of uncertainty in our sb1 estimate of β1 Confidence Intervals for and b b1 0 ➔ ➔ ➔ ➔ ➔ n small → bad big → bad se small→ bad (wants x’s spread out for s2 x better guess) Regression Hypothesis Testing *always a two­sided test ➔ want to test whether slope ( ) is β1 needed in our model ➔ : = 0 (don’t need x) H0 β1 : 0 (need x) Ha = β1 / ➔ Need X in the model if: a. 0 isn’t in the confidence interval b. t > 1.96 c. P­value < 0.05 Test Statistic for Slope/Y­intercept ➔ can only be used if n>30 ➔ if n < 30, use p­values
  • 11.
    Multiple Regression ➔ ➔ VariableImportance: ◆ higher t­value, lower p­value = variable is more important ◆ lower t­value, higher p­value = variable is less important (or not needed) Adjusted R­squared ➔ k = # of X’s ➔ Adj. R­squared will as you add junk x variables ➔ Adj. R­squared will only if the x you add in is very useful ➔ **want Adj. R­squared to go up and Se low for better model The Overall F Test ➔ Always want to reject F test (reject null hypothesis) ➔ Look at p­value (if < 0.05, reject null) ➔ : (don’t H0 ... β1 = β2 = β3 = βk = 0 need any X’s) : (need at Ha ... = β1 = β2 = β3 = βk / 0 least 1 X) ➔ If no x variables needed, then SSR=0 and SST=SSE Modeling Regression Backward Stepwise Regression 1. Start will all variables in the model 2. at each step, delete the least important variable based on largest p­value above 0.05 3. stop when you can’t delete anymore ➔ Will see Adj. R­squared and Se Dummy Variables ➔ An indicator variable that takes on a value of 0 or 1, allow intercepts to change Interaction Terms ➔ allow the slopes to change ➔ interaction between 2 or more x variables that will affect the Y variable How to Create Dummy Variables (Nominal Variables) ➔ If C is the number of categories, create (C­1) dummy variables for describing the variable ➔ One category is always the “baseline”, which is included in the intercept Recoding Dummy Variables Example: How many hockey sticks sold in the summer (original equation) ockey 00 0Wtr 0Spr 0Fall h = 1 + 1 2 + 3 Write equation for how many hockey sticks sold in the winter ockey 10 0Fall 0Spri 0Summer h = 1 + 2 3 1 ➔ **always need to get same exact values from the original equation
  • 12.
    Regression Diagnostics Standardize Residuals CheckModel Assumptions ➔ Plot residuals versus Yhat ➔ Outliers ◆ Regression likes to move towards outliers (shows up as being really high) R2 ◆ want to remove outlier that is extreme in both x and y ➔ Nonlinearity (ovtest) ◆ Plotting residuals vs. fitted values will show a relationship if data is nonlinear ( also high) R2 ◆ Log transformation ­ accommodates non­linearity, reduces right skewness in the Y, eliminates heteroskedasticity ◆ **Only take log of X variable so that we can compare models. Can’t compare models if you take log of Y. ◆ Transformations cheatsheet ◆ ovtest: a significant test statistic indicates that polynomial terms should be added ◆ : H0 ata o transformation d = n : Ha ata = o transformation d / n ➔ Normality (sktest) ◆ : H0 ata ormality d = n : Ha ata = ormality d / n ◆ don’t want to reject the null hypothesis. P­value should be big ➔ Homoskedasticity (hettest) ◆ : H0 ata omoskedasticity d = h ◆ ata = omoskedasticity Ha : d / h ◆ Homoskedastic: band around the values ◆ Heteroskedastic: as x goes up, the noise goes up (no more band, fan­shaped) ◆ If heteroskedastic, fix it by logging the Y variable ◆ If heteroskedastic, fix it by making standard errors robust ➔ Multicollinearity ◆ when x variables are highly correlated with each other. ◆ > 0.9 R2 ◆ pairwise correlation > 0.9 ◆ correlate all x variables, include y variable, drop the x variable that is less correlated to y Summary of Regression Output Source: http://people.fas.harvard.edu/~mparzen/stat100/Stat%20100%20Final%20Cheat%20Sheets%20Docs%20(2).pdf