# Exam II Review. CEE 3710 November 15, /16/2017. EXAM II Friday, November 17, in class. Open book and open notes.

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1 Exam II Review CEE 3710 November 15, 017 EXAM II Friday, November 17, i class. Ope book ad ope otes. Focus o material covered i Homeworks #5 #8, Note Packets #

2 Exam II Topics **Will emphasize material covered sice Exam I, but material is cumulative Key topics from Exam I: Uderstad basic cocepts of probability, radom variables Be able to compute percetiles Be able to use stadard ormal tables Be able to evaluate expectatios, variace for give pdf **Homework 6 Topics NOT o Exam II: No Bayes Theorem or Law of Total Probability No Biomial Distributio Distributios D. Normal Distributio (Note Packet #9) **Homework 5 Problem 5 D4. Be able to state the Cetral Limit Theorem ad kow whe it applies. (Note Packet #15) **Homework 7 Problems ad 3

3 Distributios (Note Packet #10) D5. Be able to calculate momets ad percetiles of a logormal distributio from its parameters (the log space mea ad variace µ Y, σ Y ). Be able to compute parameters give the mea ad variace of the distributio, or some mixture of percetiles ad momets. **Homework 5 Problems 1, 3, ad 6 Logormal Distributio (Packet #10) Real Space Momets (for variable of iterest X) 1 X exp Y Y X X exp Y 1 Log Space Momets (Parameters) Percetiles: l 1 1/ Y X X 1 l[ ] Y X Y x exp z p Y p Y [for Y = l(x)] 3

4 Distributios (Packet #11) D6. Be able to compute parameters of the Gumbel distributio give mea ad variace. Be able to calculate meas, variaces, percetiles ad other probabilities associated with the Gumbel distributio for specified parameter values. **Homework 5 Problems ad 4 Gumbel Distributio (Packet #11) µ = E[X] = ε /α σ = Var[X] = 1.645/α F X (x) = exp{ exp[ α(x ε) ] } < x < + Percetiles: x p = ε (1/α) l[ l(p)] 4

5 Estimators E1. Studets should kow why the cocept of a estimator is importat, ad appreciate how estimators become more accurate with larger sample sizes. E. Studets should uderstad the samplig properties of estimators of the mea ad variace of a radom variable. For X 1, X,.X ~ iid(μ, σ ) X X i / i1 E[X] 1 i 1i 1 S [X X] E[S ] = σ Var[X] / Method of Momets (E3) Packet #13 Overview of Procedure: (1) Compute sample mea ( x ) ad variace (s ) **Homework 6 () Equate sample momets to populatio momets x s = σ (3) Compute distributio parameters as fuctio of sample momets (sample data) x x fx ( x ) d x fuctio(distributio parameters) X s ( x) f ( x) dx fuctio(distributio parameters) 5

6 Cofidece Itervals for Mea (Packet #17) For a give sample with average x, a (1 α)% CI for µ =E[X] is: Case I: Mea of NORMAL data with KNOWN variace σ (ay ) xz,xz / / Case II: Mea of NORMAL data with UNKNOWN variace σ ( < 30) s xt,xt /,1 /,1 s Case III: Mea of ANY data with UNKNOWN σ but LARGE ( 30) s s xz /,xz/ **Homework 7 Problems 5 ad 6; Homework 8 Problems 1,, 3, ad 5 Cofidece Itervals for Stadard Deviatio σ (Packet #17) For give sample with stadard deviatio: 1 s ( xi x) 1 i 1 For ormal data, S /σ has a Chi squared distributio with ν = 1 degrees of freedom Ad, a (1 α)% cofidece iterval for the populatio stadard deviatio is give by: S 1 1, S,1 1,1 **Homework 8 Problems 1 ad 3 6

7 Hypothesis Testig (Packets #18 19) H1. Kow how choice of hypotheses relates to type I ad II errors ( ad β) ad be able to articulate what these errors represet. H. Kow how to select the ull ad alterative hypotheses to achieve the iteded purpose of a test, ad thus establish upo which hypothesis the burde of proof should fall (H a ), ad which will be accepted if little data is available (H o ) [i.e. usafe util prove otherwise]. H3. Be able to defie the rejectio regio for a specified α, calculate the correspodig β, ad make a decisio for oe sample test o the mea. For lower tail test (H a : μ < μ o ) **Homework 8 Problem 4 f(x Ha) f(x Ho) c Goodess of Fit Aalysis (Packet #14) G1. Studets should kow how to costruct a empirical CDF. G.Be able to costruct both probability plots ad quatilequatile plots, ad be able to idicate how such a plot illustrates whether the data is draw from the postulated distributio (e.g., Normal, logormal, Gumbel). **Homework 6; Homework 7 Problem 1; Homework 8 Problem 5 NOT ON EXAM II WILL BE ON FINAL EXAM 7

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