Stats Sheet Prelim 1 - -Samples produce different b0 and b1...

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-Samples produce different b 0 and b 1 , but sampling distribution model of regression slope is centered at β1 (the slope of idealized regression line) -Standardize Slopes by subtracting the model mean and dividing by SE Student’s t with n-2 df t (df = n – 2) = (b 1 - β1) / SE (b 1 ) Usual H0: β1 = 0, because if slope = 0, there is no linear association btwn 2 variables CI for regression slope : b 1 ± t* (df = n – 2) x SE (b 1 ) ***Regression estimates Rate of Change—CANNOT TELL CAUSATION *** SE increases, t-score decreases (less significant results) -Very low P-value means association you see in the data is unlikely to have occurred by chance reject H0 -Can Also predict mean y value for all cases OR y-value for a particular case **MORE PRECISION PREDICTING MEANS (difference is all in SE— the farther from center of our data, the LESS precise— SE for INDIVIDUAL predicted value is LARGER than SE for MEAN —extra variability) Predicting for new individual (not part of original data set) “x sub new” = xv ŷ v = b 0 + b 1 x v CI for mean predicted value : ŷv ± t* (df = n – 2) x SE (μv) “mean y value for all with that value for x” Narrower CI and smaller SE Prediction Interval for individual : ŷv ± t* (df = n – 2) x SE (ŷv) “exact y value for particular individual with that x” Wider CI and larger SE A CI—has 95% chance of capturing the true y value of a randomly selected individual with the given x-value Watch OUT 1. High influence points & outliers 2. Extrapolation 3. Make sure errors are Normal 4. Watch out for plot thickening 5. Don’t fit linear regression to data that aren’t straight Sample (statistics): Latin Letters ybar Mean μ Population (parameters): Greek Letters S Stand Dev σ R Correlation ρ Phat proportion p Categorical Data – Frequency Tables, Bar & Pie charts, Contingency Tables Quantitative Data – Histograms, Stem & leaf, dot plot, boxplot, scatterplots Marginal Distribution –distribution of either variable alone; also the counts or percentages are the totals found in the margins (last row / column) of table Data – information w/ a context (Who, what W’s of data / When, where, Why good to have) Who – called “ cases MAKE A PICTURE w/ data Relative Frequencies / Proportions depend on whether taken from column total, row total, or grand total (marginal total) 5 Number S ummary – min, max, Q1, Q3, median 4 Measures of Spread Standard Dev, IQR Measures of Position Mean, Median, Quartiles Independence – in contingency table, when the distribution of one variable is the same for all categories of another Simpson’s paradox – when averages are taken across different groups (not related), they may appear contradictory. CHANGING CENTER adding a constant to each value adds same amount to Mean, Median, and Quartiles, but DOES NOT change Stand Dev or IQR CHANGING SCALE multiplying each data value by a constant changes the measures of position (Mean, Median, Quartiles), and measures of spread too HISTOGRAMS: Each column in histogram is a bin —represents a case 1. Shape (General Trend) – a. Unimodal / Bimodal / Multimodal
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