z is the upper tail critical value from the normal distribution

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1 Statistical Iferece drawig coclusios about a populatio parameter, based o a sample estimate. Populatio: GRE results for a ew eam format o the quatitative sectio Sample: =30 test scores Populatio Samplig Dist. for X shape ormal? appro. ormal mea μ (ukow) μ 100 SD =100 (assume kow) 5.8 X (1-α)% Normal Cofidece Iterval for μ z / where / e: GRE scores (=30, σ=100, 50 ) z is the upper tail critical value from the ormal distributio % CI for μ: , (1-α)% Studet s T Cofidece Iterval for μ s t / where / t is the upper tail T critical value with -1 d.f.

2 Hypothesis Testig Null Hypothesis (H o ) vs. Alterative Hypothesis (H A ) α = P(Type I Error) the level of sigificace (l.o.s.) β = P(Type II Error) power = 1 β p-value Measures the stregth of the sample evidece agaist H o Defiitio: The probability, computed assumig that H o is true, of a sample result ( X ) as etreme or more etreme tha the oe from our sample. Rule of Thumb for the sigificace of p-values The p-value is the smallest l.o.s. α at which we ca reject H o If the p-value is less tha.05, the our results are statistically sigificat at the.05 level

3 Let X be a R.V. deotig the legth of a fish i cm. Populatio μ = 54cm?? σ = 4.5cm SRS of =9 = 57cm eperimet Meta-eperimet Does = 57 give strog evidece that μ>54 cm? How likely are we to get = 57 if μ=54 cm?

4 H o : µ=54 cm 4.5 cm σ = = 1.5 cm X 9 H a : µ>54 (µ a = 58) α =.05 4 Steps for fidig the Power i a test of hypotheses 1) Write the RR for H o i terms of z-scores: ) Write the RR for H o i terms of X : 3) Fid the probability of a Type II error if µ=58 4) Power = 1 P(Type II Error) :

5 H o : µ=54 cm 4.5 cm σ = = 1.5 cm X 9 H a : µ>54 (µ a = 58) α =.05 4 Steps for fidig the Power i a test of hypotheses 1) Write the RR for H o i terms of z-scores: zs ) Write the RR for H o i terms of X : X X ) Fid the probability of a Type II error if µ=58 β(58) = P(Accept H o H a is true [µ=58] ) 4) Power = 1 P(Type II Error) : PWR(58) = 1 β(58)

6 Probability Distributios for Radom Variables (RVs) Probability Distributio (for a discrete RV) represeted graphically as a Probability Histogram - Idicates the possible values for a RV - Idicates how to assig probabilities for the possible values: p() = P( X = ) Probability Distributio (for a cotiuous RV) represeted as a Probability Desity Curve - Areas uder a smooth curve, f(), idicate probabilities of values i a give rage Epected Value of a Radom Variable a weighted average of all possible values for X, weighted by the probability of each value E( X) = μ, the mea for the RV o E( X) = p( ) for a discrete RV = o E( X) = f( ) d for a cotiuous RV = Eample (discrete RV): Y = # heads i two tosses of a fair coi EY ( ) = 0 ipy ( = 0) + 1 ipy ( = 1) + ipy ( = ) = 0i + 1i + i = P( Y = 0 ) = ¼ P( Y = 1 ) = ½ P( Y = ) = ¼ ad zero otherwise Eample (cotiuous RV): X = the positio at which a two-meter with legth of rope breaks whe put uder tesio (assumig every poit is equally likely) f()= ½ 0, zero otherwise f() 1 1/ 1 1 ( ) = = = 4 = 0 EX d Properties of Epectatio E(a) =a E(aX) =a*e(x) E(X+a)= E(X) + a E(X+Y)= E(X) + E(Y) E(XY) = E(X) * E(Y) if X & Y are idepedet

7 Variace of a Radom Variable a weighted average of squared deviatios from the mea, [ E()] Var( X ) = E[( X μ) ] = σ o o ( ) Var( X ) = E( ) p( ) = ( ) Var( X ) = E( ) f ( ) d = for a discrete RV for a cotiuous RV Var X E X E X E X ( ) = [( μ) ] ( ) [ ( )] for ay RV Eample: Y = # heads i two tosses of a fair coi OR P( Y = 0 ) = ¼ P( Y = 1 ) = ½ μ = EY ( ) = 1 P( Y = ) = ¼ ad zero otherwise ( ) = (0 μ) ( = 0) + (1 μ) ( = 1) + ( μ) ( = VarY ipy ipy ipy = (0 1) i + (1 1) i + ( 1) i = EY PY PY PY ( ) = 0 i ( = 0) + 1 i ( = 1) + i ( = = 0i + 1i + 4i = 3 1 Var( Y ) E( Y ) [ E( Y )] 1 Properties of Variace Var(X±a) = Var(X) Var(aX) =a *Var(X) = = = Var(X+Y) = Var(X) + Var(Y) if X & Y are idepedet ) ) Var(X+Y) = Var(X) + Var(Y) + *Cov(X,Y) always Cov(X,Y) = E( [X E(X)] * [Y E(Y)] ) = E( XY ) E(X)*E(Y)

8 iid Y, Y,, Y N( μ, σ ) a Simple Radom Sample (SRS) of size 1 i.i.d. idepedet ad idetically distributed Y1+ Y Y ( ) ( 1 ) 1 EY = E = E Y+ Y + + Y = ( i μ) = μ Y1+ Y + + Y 1 1 σ Var( Y ) = Var = Var ( Y1+ Y + + Y) = ( i Var( Yi) ) = Y μ = Z σ N(0,1) -Sample Tests iid 11, 1,, 1 ( μ1, σ1 ) 0 : μ1 = μ Y Y Y N H 1 iid 1 μ σ Y, Y,, Y N(, ) EY ( Y) = EY ( ) EY ( ) = μ μ σ σ Var( Y1 Y) = Var( Y1) + Var( Y) = + 1 1

9 Ho: μ=165 mg/dl Ha: μ>165 Sample Results: A SRS of = 30 gives X = = 41.3 (assume this is kow) a) Fid the sample z-score (zs ). [uses 180.5] b) Fid the p-value. c) State a coclusio for the test at the α =.05 level. 1) Write the rejectio rule (RR) for Ho i terms of z-scores. [does NOT use 180.5] ) Write the rejectio rule (RR) for Ho i terms of X. 3) Fid the probability of a Type II error if \mu=170.

10 Four Key Probability Distributios 1. Normal Distributio (assumig is a kow populatio parameter) o CLT Samplig Distributio for X is approimately N, X o zs has a N(0,1) distributio /. Studet s T Distributio (assumig is ukow) ( i ) o we estimate with s 1 X o ts has a Studet s T distributio with -1 degrees of freedom s/ o Stadard Deviatio (of the mea) vs. Stadard Error (of the mea) s SD X SE X Normal Distributio i blue T Distributio df = 5 i gree T Distributio df = 10 i red

11 3. Chi-Square Distributio o Variaces follow a scaled Chi-Square Distributio Y iid i o zi N(0,1) o z z 1 Yi i 1 z zi i1 i1 ( Yi ) ~ o Estimatig μ with ˆ Y i1 ( Yi Y) ~ 1 S 1 ( Y i Y) ( 1) S ~ 1 4. F Distributio o A ratio of two scaled Chi-square radom variables o ( 1) S ~ 1 S S 11 1 o S * F * uder H : 1 S , 1 o 1 o If T is Studet s T radom variable with m df, the T F 1, m