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# PRIN OF STATISTICS I STAT 211

Texas A&M

GPA 3.5

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This 7 page Class Notes was uploaded by Celestino Bergnaum on Wednesday October 21, 2015. The Class Notes belongs to STAT 211 at Texas A&M University taught by Erning Li in Fall. Since its upload, it has received 289 views. For similar materials see /class/225746/stat-211-texas-a-m-university in Statistics at Texas A&M University.

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Date Created: 10/21/15

Stat381 Cheat sheet Confidence Intervals Parameters being Begin Derivation lOO1a estimated with Distribution DF conf interval 2 Y u u a known Z Xiza a J aJ 2 Y u 02 unknown T Xita S J SJ n 1 2 1 2 1 2 0392 n 1SZO2 11 lewjljzzw lac2 Ilia2 P proportion amp N Z iZan U n 12 1122 W Z fl Y2iZa2O 12nl i a22n2 01702 known 2 2 I 2 I X1 X2 1 2 Tn in 2 Z mi rm 520111 1112 unknown amp equal S 127 1 n1 1n2 1 2 Si n1 1S12 n2 1S22 p r11 H2 2 Y Lil 2 0127022 W Ty X1X2iZa2VSIZnlS n2 unknown amp unequal 51 n1 52712 y 512 n1 SZZn22Slz 7627 Z n1 1 n2 1 2 2 2 2 0102 SI S2 Fnlimrl Liz17a 2aL2 2ft 2 2008 Xijin Ge All rights NOT reserved Accuracy NOT guaranteed Note PT gt tan 2 012 RightTailed 539 Two tailed H Ho39L1qu 1 1045qu H1LL1gtuZ H1u1 uZ t It Jt I r v j r W W Reject Ha Reject Ha Reject Ha Accept Ha Accept Ha Reject Ha Accept Hg P value PUT 2 V j P 1 PTT P a1 PT2T W W e WW1 V e Md mp mm lt o 2X1 T 2mm imam gt0 Quick recipe 1 One sample T test Hypothesis testing on the mean General procedure to test the hypothesis that Ho 1 Mo when you have observed n values from a normal population If you know the variance oz 1 Calculate sample mean 2 Calculate the statistic Z y o aJZ 3 At 5 significance level accept Ho if Z is between 196 and 196 4 Use normal dist table to determine critical point at other significance levels Z If you don t known the variance 1 Calculate sample mean X 2 Calculate sample variance 52 S2 X12X22Xj n72 11 1 3 Calculate thestatistic T T X IUO 54 4 The Tstatistic will have a T dist with n 1 degree of freedom if the null hypothesis is true Make decision based on table in the context of left right or two tailed test Reference Downing amp Clark Statistics the easy way Barron s Educational Series 1997 Original formula for sample variance n1 SX1 22X 22Xif Quick recipe 2 comparing two means with known variances rarely used Hypothesis testing on the difference between two means known variances Null hypothesis Ho 11 2 0 D is often zero If variances 03912 022 known 1 Calculate X1 and X2 2 Calculate the test statistic Z 1 X2 D 012171 022n2 3 Z is normally distributed At a 005 level if Z is between l96 and 196 accept H0 u1 u2 D 4 Use normal distributation table to determine the P value or critical values at another signi cance level 4 Quick recipe 3 T test with equal variances Hypothesis testing on the difference between two means egual variances Null hypothesis Ho J1 2 0 D is often zero If variances 0 12 0 22 are unknown but assumed to be equal 1 Calculate X1 and X2 2 Calculate the sample variances SI2 S 22 with n 1 in the denominator 3 Estimate the pooled variance S n1 1S12 112 1S22 111 112 2 4 Calculate the test statistic T T 2 X1 X1 D 1liS 1nl1nz 5 If the null hypothesis is true then T will have a T distribution Si with 111 112 2 degrees of freedom Use T table to determine critical values at a given signi cance level or P value for observed T In Excel TTESTarray1array2tails2 Quick recipe 4 T test with unequal variances Hypothesis testing on the difference between two means unegual variances Null hypothesis Hozu1 2 0 D is often zero 2 2 Normality required If Varlances 01 a 02 are unknown and unequal Safe to use even variances equal lCalculate Y1 and 2 2 Calculate the sample variances Sf 22 with n l in the denominator 3 Calculate the Smith Satterthwaite degrees of freedom 7 Slznl SZZn22 Sf n12 Sinzf n1 1 n2 1 4 Calculate the test statistic T T Z 1 D I EXCEL TTEST 1 2 I 3 n array array tai s l512n1522n2 5 If the null hypothesis is true then T will have a T distribution with 7 degrees of freedom Use T table to determine critical value at a given significance levelthe P value for observed T Quick recipe 5 Paired T test Hypothesis testing on the difference between two means Paired data Null hypothesis Ho luX luY 0 Normality required If two random samples X Y are pa1red 1 For each paired observation X 1 K where i 12 n calculate the difference DZ Di Xi 2 We have de ned a new random variable D and converted a two sample problem into a one sample problem The null hypothesis is 1D 2 O 3 Calculate the sample mean 5 and variances S 12 2 2 2 2 5 D1 12 Dn7 and Sf D1 12 Dn nD n n l 4 Calculate the test statistic D T SD NZ 5 If the null hypothesis is true then T will have a T distribution with n 1 degrees of freedom Use T table to determine critical value at a given signi cance level or P Value for Observed T39 In EXCEL TTESTarray1array2tails1

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