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# PRIN OF STATISTICS I STAT 211

Texas A&M

GPA 3.54

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This 4 page Class Notes was uploaded by Darien Kutch on Wednesday October 21, 2015. The Class Notes belongs to STAT 211 at Texas A&M University taught by Samiran Sinha in Fall. Since its upload, it has received 35 views. For similar materials see /class/225750/stat-211-texas-a-m-university in Statistics at Texas A&M University.

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Date Created: 10/21/15

Crib Sheet for Exam 2 Statistics 211 1 Chapter 5 Joint Probability Distributions Marginal probability massdensity function we 72pm pyltygt72pltmgt mac 7 mm my mm Y X 700 700 TWO RV X Y are independent if for every pair of X Y 1017 9 PX I Pyy ay 121 My otherwise they are dependent Conditional probability density function of Y given that X z is ay fyix f 7 00 lt y lt oo pY X May 1 I Expected values Ewangt1Zmepm Emmigt17 m mhltzygtfltzygtdzdy Gamay 7 EX 7 my 7 m 7 Em 7 My may COTTXY w 71S pg S 1 aXaY Central Limit Theorem CLT states that if our random variables X have a dis tribution With Whose mean and variance exist and X1 X2 Xn are a random sample and our sample is large say n 2 30 then 2 7 a X N 1W7 2 Linear combinations Ea1X1 aan a1EX1 anEXn Va1X1 aan ZZaiajC0vXZ Xj i1 j1 lf X1 Xn are mutually independent then Va1X1 aan a VX1 ailan since C0vXZXj 0 When ia j and C0vXZXi VXi lf X1 Xn are independent and normal though they may have different M and al for each Xi then any linear combination of X s also has a normal distribution 2 Chapter 7 Confidence Intervals Basic con dence interval7 a is known 7 a z i Zed2W Large sample Cl 7 s z i Zed2W Cl for p if n13 2 5 and m 2 5 and n is small compared to the population size N 5 i Z042 lln A small sample Cl 3 i i tut27171 l i i tut27L71 8H1 TL n7 1 n7 1 J 2 7 2 XaQmil Mia27171 Prediction interval Cl for 02 Sample size calculations a 2 4Z2 t n lt2Za2Zgt n L22 3 Chapter 8 One Sample Hypothesis Tests One Sample test for mean 7 Normal distribution amp a is known i i 0 Z a Alternative hypothesis Rejection region for level a test Ha M gt 0 Z 2 Z0 Ha I M lt 0 Z S 7Z0 Hail flio ZZZa2 0r ZS Za2 Type ll Error Formulas for 50 Alternative hypothesis Type ll error for size a test H 7M HaAMgtMo ltIgtlt2aU N gt H 7M Hazlt0 li lt7za a gt HaMH Mo at gm i a wt2 m Sample sizes and Type ll errors Alternative hypothesis Sample size Hazwm HaiMiMo Wf One Sample Test for Mean 7 Large Sample CLT iiiL0 Z s Rejection region is the same as one sample Ztest abovei One sample test for mean 7 Normal distribution amp a is unknown film 5N5 Alternative hypothesis Rejection region for level a test Ha M gt 0 75275057171 Ha M lt 0 75S 45047171 Ha 1 i 0 752 tut27L71 OF 75S itaQmil Large sample tests concerning a population proportion p Alternative hypothesis a I p gt p0 Ha I P lt P0 Ha P Po P PO P0 0n Rejection region for level a test Z gt Z Z E 720 Z 2 Z042 or Z S Za2 Note that npo 2 5 and ram 2 5 for this test to hold Type ll error formulas for Bp Alternative hypothesis Ha I P gt P0 Ha I P lt P0 HarPy Po Type ll error for size a test PeepZMpoqon PWn 17 i PoiPizaxPoqongt PHIM ltPoiPZa2P0110n xPqhl 7 PeepZazlfpoqon xPqM Sample sizes and Type ll errors Alternative hypothesis Sample size Ha I gt P0 JAN t pull 2 Ha I lt P0 MM t pull 2 Ha p p0 2 The pValue for a Z test is calculated as follows 17 ltIgtZ for an uppertailed test for a lowertailed test 2l 7 for a twotailed test pValue

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