Review of Statistics
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1 Review of Statistics
2 Topics Descriptive Statistics Mean, Variance Probability Union event, joint event Random Variables Discrete and Continuous Distributions, Moments Two Random Variables Covariance and correlation Central Limit Theorem Hypothesis testing z-test, p-value Simple Linear Regression
3 Statistical Methods Statistical Methods Descriptive Statistics Inferential Statistics
4 Descriptive Statistics Involves Collecting Data Presenting Data Characterizing Data Purpose Describe Data st Qtr 2nd Qtr 3rd Qtr 4th Qtr East West North
5 Inferential Statistics Involves Estimation Hypothesis Testing Population? Purpose Make Decisions About Population Characteristics
6 Descriptive Statistics
7 Mean Measure of central tendency Acts as Balance Point Affected by extreme values ( outliers ) Formula: X n Xi i = = = n X + X + + X n n
8 Median Measure of central tendency Middle value in ordered sequence If odd n, Middle Value of Sequence If even n, Average of 2 Middle Values Value that splits the distribution into two halves Not Affected by Extreme Values
9 Median (Example) Raw Data: Ordered: Position: Median = = 16
10 Mode Measure of Central Tendency Value That Occurs Most Often Not Affected by Extreme Values There May Be Several Modes Raw Data: Ordered:
11 Sample Variance S 2 = n i = 1 (X X) i n 1 2 n - 1 in denominator! (Use n if population variance) = (X X) + (X X) + + (X X) n n
12 Sample Standard Deviation S = S 2 = n i = 1 (X X ) i n 1 2 = (X X ) + (X X ) + + (X X ) n n
13 Probability
14 Event, Sample Space Event: one possible outcome Sample space: collection of all the possible events S = { } Probability of an outcome: proportion of times that the outcome occurs in the long run The complement of event A: includes all the events that are not part of the event A: Symbol A Event A { } Complement of A A { }
15 Properties of an Event 1. Mutually Exclusive Two outcomes that cannot occur at the same time Experiment: Observe gender of one person 2. Collectively Exhaustive One outcome in sample space must occur
16 Joint Events Joint event: Event that has two or more characteristics means intersection of event (set) A and event (set) B Example: A and B, (A B): Female, Under age 20
17 Compound Events Union of event A and event B ( A B ): Total area of the two circles A B contains all the outcomes which are part of event (set) A, part of event (set) B or part of both A and B means union of event A and event B
18 Compound Probability Addition Rule Used to Get Compound Probabilities for Unions of Events P(A OR B) = P(A B) = P(A) + P(B) - P(A B) For Mutually Exclusive Events: P(A OR B) = P(A B) = P(A) + P(B) Mutually Exclusive Events A B
19 Random variables Random variable numerical summary of a random outcome a function that assigns a numerical value to each simple event in a sample space Discrete or continuous random variables Discrete: only a discrete set of possible values => summarized by probability distribution: list of all possible values of the variables and the probability that each value will occur. Continuous: continuum of possible values => summarized by the probability density function (pdf)
20 Discrete Probability Distribution 1. List of pairs [ X i, P(X i ) ] X i = Value of Random Variable (Outcome) P(X i ) = Probability Associated with Value 2. Mutually exclusive (no overlap) 3. Collectively exhaustive (nothing left out) 4. 0 P(X i ) 1 5. Σ P(X i ) = 1
21 Joint Probability Using Contingency Table Event Event B 1 B 2 Total A 1 P(A1 B1) P(A1 B2) P(A 1 ) A 2 P(A2 B1) P(A2 B2) P(A2) Total P(B 1 ) P(B 2 ) 1 Conditional probability: P( AB1) P( A1 B1) PB ( 1) PA ( 2 B1) PB ( ) 1 1 Joint Probability Joint distribution: Marginal Probability Marginal distributions: Conditional distribution:
22 Contingency Table Example Joint Event: Draw 1 Card. Note Kind, Color Color Type Red Black Total Ace 2/52 2/52 4/52 Non-Ace 24/52 24/52 48/52 Total 26/52 26/52 52/52 P(Ace) P(Red) P(Ace AND Red)
23 Moments Discrete Case Moment: Summary of a certain aspect of a distribution Mean, Expected Value Mean of Probability Distribution Weighted Average of All Possible Values μ = E(X) = ΣX i P(X i ) Variance Weighted Average Squared Deviation about Mean σ 2 = E[ (X i μ) 2 ] = Σ (X i μ) 2 P(X i )
24 Statistical Independence When the outcome of one event (B) does not affect the probability of occurrence of another event (A), the events A and B are said to be statistically independent. Example: Toss a coin twice => no causality Condition for independence: Two events A and B are statistically independent if and only if (iff) P(A B) = P(A)
25 Bayes Theorem and Multiplication Rule Bayes Theorem P(A B) = P(A B) P(B) The difficult part is P(A B) Use above equation to derive P(A B) P(A and B) = P(A B) = P(A)P(B A) = P(B)P(A B) For independent events: P(A and B) = P(A B) = P(A)P(B)
26 Covariance Measures the joint variability of two random variables N σ XY = (X i μ X )(Y i μ Y )P(X i, Y i ) i=1 Can take any value in the real numbers Depends on units of measurement (e.g., dollars, cents, billions of dollars) Example: positive covariance = y and x are positively related; when y is above its mean, x tends to be above its mean; when y is below its mean, x tends to be below its mean.
27 Correlation Standardized covariance, takes values in [-1, 1] Does not depend on unit of measurement Correlation coefficient (ρ) formula: ρ = cov( XY ) σ σ X Y = σ σ X XY σ Covariance and correlation measure only linear dependence! Example: Cov(X,Y)=0 Does not necessarily imply that y and x are independent. They may be non-linearly related. But if X and Y are jointly normally distributed, then they are independent. Y
28 Sum of Two Random Variables Expected Value of the Sum of Two Random Variables E(X + Y) = E(X) +E(Y) Variance of the Sum of Two Random Variables Var (X + Y) = σ 2 = + σ 2 Y + 2σ XY σ 2 X X+Y
29 Continuous Probability Distributions - Normal Distribution Bell-Shaped, symmetrical Mean, median, mode are equal Infinite range 68% of the data are within 1 standard deviation of the mean 95% of the data are within 2 standard deviations of the mean In early 1800's, German mathematician and physicist Karl Gauss used it to analyze astronomical data, therefore known as Gaussian distribution. f(x) Mean, Median, Mode X
30 Normal Distribution Probability Density Function f( X) = 1 2 e π σ 1 2 X μ σ 2 f(x) = frequency of random variable X π = ; e = σ = population standard deviation X = value of random variable (- < X < ) μ = population mean
31 Effect of Varying Parameters (μ & σ) f(x) B A C X
32 Normal Distribution Probability Probability is the area under the curve! d Pc ( X d) = f ( x) dx? c f(x) c d X
33 Infinite Number of Normal Distribution Tables Normal distributions differ by mean & standard deviation. Each distribution would require its own table. f(x) X That s an infinite number!
34 Standardize the Normal Distribution Normal Distribution σ Z = X μ σ Standardized Normal Distribution σ z = 1 μ X μ Z = 0 Z One table!
35 Standardizing Example Normal Distribution σ = 10 Z = X μ = = 0.12 σ 10 Standardized Normal Distribution σ Z = 1 μ = X μ Z = 0.12 Z
36 Moments: Mean, Variance (Continuous Case) Mean, Expected Value Mean of probability distribution Weighted average of all possible values Variance μ = E(X) = X f(x) dx - Weighted average squared deviation about mean σ 2 = E[ (X μ) 2 ] = (X- μ) 2 f(x) dx -
37 Moments: Skewness, Kurtosis ( ) 3 Skewness: E X μ S = 3 Measures asymmetry in distribution σ The larger the absolute size of the skewness, the more asymmetric is distribution. A large positive value indicates a long right tail, and a large negative value indicates a long left tail. A zero value indicates symmetry around the mean. ( μ ) 4 E X Kurtosis: 4 σ Measures thickness of tails of a distribution A kurtosis above three indicates fat tails or leptokurtosis, relative to the normal, i.e. extreme events are more likely to occur. K =
38 Central Limit Theorem: Basic Idea As sample size gets large (n 30)... sample mean will have a normal distribution. X
39 Important Continuous Distributions All derived from normal distribution 2 χ distribution: arises from squared normal random variables, t distribution: arises from ratios of normal 2 and χ variables F distribution: arises 2 from ratios of χ variables. 2 χ t distribution (red), normal distribution (blue) distribution F distribution
40 Fundamentals of Hypothesis Testing
41 Identifying Hypotheses 1. Question, e.g. test that the population mean is equal to 3 2. State the question statistically (H 0 : μ = 3) 3. State its opposite statistically (H 1 : μ 3) Hypotheses are mutually exclusive & exhaustive Sometimes it is easier to form the alternative hypothesis first. 4. Choose level of significance α Typical values are 0.01, 0.05, 0.10 Rejection region of sampling distribution: the unlikely values of sample statistic if null hypothesis is true
42 Identifying Hypotheses: Examples 1. Is the population average amount of TV viewing 12 hours? μ = 12 μ 12 H 0 : μ = 12 H 1 : μ Is the population average amount of TV viewing different from 12 hours? μ 12 μ = 12 H 0 : μ = 12 H 1 : μ 12
43 Hypothesis Testing: Basic Idea Sampling Distribution It is unlikely that we would get a sample mean of this value if in fact this were the population mean.... Therefore, we reject the null hypothesis that μ = μ = 50 H 0 sample mean
44 Example: Z-test statistic (σ known) 1. Convert Sample Statistic (e.g., ) to Standardized Z Variable X μx X μ Z = = σ σ x n 2. Compare to Critical Z Values If Z-test statistic falls in critical region, reject H 0 ; Otherwise do not reject H 0 X
45 p-value Probability of obtaining a test statistic more extreme ( or ) than actual sample value given H 0 is true Smallest value of α for which H 0 can be rejected Used to make rejection decision If p value α, do not reject H 0 If p value < α, reject H 0
46 One-Tailed Test: Rejection Region H 0 : μ 0 H 1 : μ < 0 H 0 : μ 0 H 1 : μ > 0 Reject H 0 α Reject H 0 α 0 Must be significantly below μ. Z 0 Z Here: Small values don t contradict H 0.
47 One-Tailed Z Test: Finding Critical Z Values What Is Z Given α = 0.025? σ Z = Z α /2 =.025 Standardized Normal Probability Table (Portion).06 Z
48 Two-Tailed Test: Rejection Regions Sampling Distribution Rejection Region 1/2 α 1 - α Nonrejection Region H 0 : μ = 0 H 1 : μ 0 Level of Confidence Rejection Region 1/2 α Critical Value H 0 Value Critical Value Sample Statistic
49 t-test, F-test Test statistic may not be normally distributed => z-test not applicable Examples: Variance unknown, but estimated. Hypothesis that the slope of a regression line differs significantly from zero. => t-test Hypothesis that the standard deviations of two normally distributed populations are equal. => F-test
50 Jarque-Bera test Assesses whether a given sample of data is normally distributed. Aggregates information in the data about both skewness and kurtosis. Test of the hypothesis that S = 0 and K = 3, based on Ŝ and ˆK. Test statistic: T ˆ 2 1 ( ˆ ) 2 JB = S + K (here T is the number of observations) Under the null hypothesis of independent normallydistributed observations, the Jarque-Bera statistic is 2 distributed in large samples as a χ random variable with 2 degrees of freedom.
51 Simple Linear Regression
52 Simple Linear Regression Model y-intercept slope random iid 2 error ε (0, σ ) t y = + x + t 0 1 t t β β ε dependent (response) variable independent (explanatory) variable
53 Linear Regression Assumptions 1. x is exogenously determined 2. ε t are iid(0,σ 2 ) (iid = independently and identically distributed ) Zero mean Independence of errors (no autocorrelation) Constant variance (homoscedasticity) More things to think about: Normality of ε t (if not satisfied, inference procedures only asymptotically valid) Model specification (e.g. linearity, β 1 constant over time?)
54 Simple Linear Regression Model y y = β + β x + ε t 0 1 t t ε t = disturbance observed value observed value ( ) E yx = β + β x * * 0 1 x
55 -- Sample Linear Regression Model y y = b + b x + e i 0 1 i i e i = Random Error y = b + b x i 0 1 i Unsampled Observation Observed Value x
56 Ordinary Least Squares OLS minimizes sum of squared residuals min ˆ β, ˆ β 0 1 y T t= 1 ( y ˆ ˆ ) t β0 β1xt 2 y = β + β x + ε t 0 1 t t T ( ) 2 2 y yˆ = e t t t t= 1 t= 1 T predicted value e 2 e 4 e 1 e 3 fitted value (in-sample forecast) y = ˆ β + ˆ β x ˆt 0 1 x t
57 On Thursday: Evaluating the Model 1. Examine variation measures coefficient of determination ( goodness of fit ) standard error of the estimate 2. Analyze residuals e serial correlation 3. Test coefficients for significance β y ˆt = ˆ β + ˆ β x 0 1 t
58 Random Error Variation 1.Variation of Actual Y from Predicted Y 2. Measured by Standard Error of Estimate Sample Standard Deviation of e Denoted S YX 3. Affects Several Factors Parameter Significance Prediction Accuracy
59 Measures of Variation in Regression 1.Total Sum of Squares (SST) Measures variation of observed Y i around the mean, Y 2.Explained Variation (SSR) Variation due to relationship between X & Y 3.Unexplained Variation (SSE) Variation due to other factors
60 Variation Measures Y Y i Unexplained Sum of Squares (Y i - Y ^ i ) 2 Total Sum of Squares (Y i -Y) 2 X i Explained Sum of Squares (Y ^ i -Y) 2 Y = b + b X i Y 0 1 X i
61 Coefficient of Determination Proportion of Variation Explained by Relationship Between X & Y r 2 Explained Variation SSR = = Total Variation SST = n n b Y + b XY n(y) 0 i 1 i i i = 1 i = 1 n 2 Yi i = 1 n(y) 2 ˆ 2 0 r 2 1
62 Coefficients of Determination (r2) and Correlation (r) Yr 2 = 1, r = +1 Y r 2 = 1, r = -1 ^ Y^ i = b 0 + b 1 X i X Y i = b 0 + b 1 X i Yr 2 =.8, r = +0.9 Y r 2 = 0, r = 0 X Y^ i = b 0 + b 1 X i X Y^ i = b 0 + b 1 X i X
63 Standard Error of Estimate 2 2 ) ˆ ( = = = = = = n Y X b Y b Y S n Y Y S n i n i n i i i i i YX n i i i YX
64 Residual Analysis 1.Graphical Analysis of Residuals Plot residuals vs. X i values Residuals mean errors Difference between actual Y i & predicted Y i 2.Purposes Examine functional form (linear vs. non-linear Model) Evaluate violations of assumptions
65 Test of Slope Coefficient for Significance 1.Tests If There Is a Linear Relationship Between X & Y 2.Hypotheses H 0 : β 1 = 0 (No Linear Relationship) H 1 : β 1 0 (Linear Relationship) 3.Test Statistic b = 1 t β 1 n 2 S b 1 where S = b 1 S YX n X 2 n( X ) 2 i i = 1
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