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Final

UASTAT141FinalReview.pdf

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Department
Statistics
Course
STAT141
Professor
Paul Cartledge
Semester
Winter

Description
Chapters 1-6: Population vs. sample Æ Parameter vs. statistic Statistics: Descriptive vs. inferential Types of variables Quantitative vs. Qualitative / | Discrete Continuous Tables, charts & graphs - frequency tables - qualitative: bar graph/pie chart - stem-and-leaf plot/dot plot - time plot - histogram (modality) - traits: # of modes, tail weight, overall shape (symmetry, skewness) - identify skewness by TAIL - boxplot (skewness) - outliers, overall shape (symmetry, skewness) - identify skewness inside box or entire graph Measures of center/spread/position - center: mean, median, mode Æ Outlier effect? Skewness effect? - spread: range, variance, standard deviation, IQR Æ Why use squared and (n – 1)? Ever negative? Empirical Rule? - position: min, max, percentiles (quartiles) Æ recall that we INCLUDE the median when determining quartiles Æ 5-number summary, boxplot, types of outliers Chapters 7-10: Displaying bivariate data - scatterplot: visual aid to see form/strength/direction of relationship and/or outliers (large residual, high leverage, influential) - correlation: numerical aid to see strength/direction of relationship (range?) Æ Warning: assumes linearity, sensitive to outliers Simple linear regression analysis - regression line: ŷ = b + b x 0 1 ⎛ sy ⎞ - least-squares estimation gives b1 = r⎜ ⎟ and b 0 y −b x 1 ⎝ sx ⎠ - estimation: interpolation vs. extrapolation (BAD!) - R-squared: r 2 = proportion of variation in y explained by x - causation: association does NOT imply causation - residual plots: observed vs. theoretical appearance - transformation of a variable can help improve linearity Chapter 11-13: - observational/retrospective/prospective study, experiment/controlled clinical trial Æ population and causal inferences (what needs to be present for each?) - types of bias (response, undercoverage, nonresponse) - types of sampling: with/without replacement, SRS/stratified/cluster/ voluntary/convenience/systematic - controlling factors: randomization, blocking, direct control, replication - more experiment design definitions Chapters 14-15: - types of events: marginal, conditional, union, intersection, complement, - What common words identify them? - relating events: dependent vs. disjoint vs. independent - Do these relations affect the rules below? If so, how? - Do they allow certain rules to be easily extended? - probability laws: - conditional probability: P(A| B) =P(A∩ B) P(B) - complement rule:P(A ) = 1 – P(A) - multiplication rule: P( A∩ B) = P(A and B) = P(A) × P(B | A) = P(B) × P(A | B) - addition rule: P(A or B) = P(A) + P(B) – P(A and B) - total probability rule:A) P= +()∩P ∩A B C ( ) - recall examples where we combined a few of these together Chapter 16-17: Distributions - discrete (exact probability or intervals) vs. continuous (only intervals) DisIcfrete: P(X = a) > 0, then P(X ≤ a) ≠ P(X < a) Continuous: If P(X = a) = P(X = b) = 0, then P(a ≤ X ≤ b) = P(a < X < b) - discrete distributions: - determine probability distribution (values of X and corresponding probabilities) - mean: µ = ∑ xiP(X = x i - variance: σ = ∑ (xi− µ) P(X = x )i= ∑ xiP(X = x ) i µ 2 n n! n ⎜ ⎟ n! - permutations/combinations: P r (n− r)! and C r= ⎜ r = r!(n −r)! ⎝ ⎠ - binomial dist’n: indep. trials, two outcomes/trial, constant p, X = # of successes ⎛ ⎞ x n−x f( ) = ⎜ ⎟ p (1− p) x = 0, 1, …, n ⎝ ⎠ µ = E(X) = np and σ =V(X))= np(1− p - continuous distributions: - uniform distribution: finding an area of a rectangle (with a twist!) - normal distribution: symmetric, 2 parameters: µ and σ, other properties Standard Normal Distribution (and its applications) - µ = 0 and σ = 1 - Table Z only gives areas to left of valuez, conversion to these values required Æ use diagrams, complements, symmetry, etc. ⎛ X − µ x− µ ⎞ - standardizing: P(X ≤ xÆ P⎜ σ ≤ σ ⎟= P(Z ≤ z) ⎝ ⎠ - identifying values for a given probability: x = µ + zσ - normal approximation to binomial: If X ~ B(n, p), np ≥ 10 and n(1 – p) ≥ 10, then ⎛ ⎞ xpn PX() ≤≈ P Z ⎜ ⎟ np (1− p ) ⎝ ⎠ Combinations and Functions of Random Variables For any constants a and b, Means: Variances: 1. E(a) =. a 1 V(a) = 0 2. E(aX) = aE(X) 2. V(aX) = a V(X) 2 3. E(aX + b) = aE(X) + b 3. V(aX + b) = a 2(X) 2 4. E(aX ± bY) = aE(X) ± bE(Y) 4. V(aX ± bY) = a V(X) + b V(Y) ± 2abcov(X, Y) Y = a1 1+ a 2 2 … + a X n n, E(Y) = a E(1 ) +1a E(2 ) +2… + a E(Xn) +nb If X1, 2 , …, n are independent,V(Y) = a 1(X ) 1 a V2X ) +2..+ a V(n ) n Chapter 18: Sampling Distributions - sample proportion: p(1− )p pq Rule 1: µ p p. Rule 2:σ p = = . n n Rule 3: If np and n(1 – p) are both ≥ 15, then has an approx. normal dist’n. ˆ All 3 rules Æ If rule 3 holdsZ N pp− ▯ (0,1) p(1− ) n - sample mean: σ Rule 1: µ y µ . Rule 2: σ y . n Rule 3: When the population distribution is normal, the sampling distribution of y is also normal for any sample size n. Rule 4 (CLT): When n > 30, the sampling distribution of y is well approximated by a normal curve, even when the population distribution is not itself normal. All 4 rules Æ If n is large OR the population is normal, N Y −µ ▯ (0,1) σ / n Chapter 19: - how to interpret CI? - generic CI: point estimate ± (critical value) × (standard error) Æ confidence level increases, ME increases Æ n increases, ME decreases - sample proportion: Assumptions: random sample, np ≥ 15 and n(1− p) ≥ 15. p(1 − p) p ± zα / 2 n - choosing n: z 2 n ≈ p (1
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