ࡱ> VXU` >Qbjbj p|)4$$$h%|%\?v% ((:(:((///? ? ? ? ? ? ?$AhC.?-0.l/00.?:(([?2220j:(L(?20?22R <p^2>(% H3G$r0=>4q?0?=f;Dl1;D<2>2>B;Dt>`//2/ /S///.?.?f2j///?0000$$ The Distribution of Sample Means Inferential statistics: Generalize from a sample to a population Statistics vs. Parameters Why? Population not often possible Limitation: Sample wont precisely reflect population Samples from same population vary sampling variability Sampling error = discrepancy between sample statistic and population parameter Extend z-scores and normal curve to SAMPLE MEANS rather than individual scores How well will a sample describe a population? What is probability of selecting a sample that has a certain mean? Sample size will be critical Larger samples are more representative Larger samples = smaller error The Distribution of Sample Means Population of 4 scores: 2 4 6 8 ( ( = 5 4 random samples (n = 2):  EMBED Equation 1= 4  EMBED Equation 3 = 5  EMBED Equation 2 = 6  EMBED Equation 4 = 3  EMBED Equation  is rarely exactly ( Most  EMBED Equation  a little bigger or smaller than ( Most  EMBED Equation  will cluster around ( Extreme low or high values of  EMBED Equation  are relatively rare With larger n,  EMBED Equation s will cluster closer to (the DSM will have smaller error, smaller variance) A Distribution of Sample Means  The Distribution of Sample Means A distribution of sample means ( EMBED Equation ) All possible random samples of size n A distribution of a statistic (not raw scores) Sampling Distribution of  EMBED Equation  Probability of getting an  EMBED Equation , given known ( and ( Important properties (1) Mean (2) Standard Deviation (3) Shape Properties of the DSM Mean? ( EMBED Equation  = ( Called expected value of  EMBED Equation   EMBED Equation  is an unbiased estimate of ( Standard Deviation? Any  EMBED Equation  can be viewed as a deviation from ( ( EMBED Equation  = Standard Error of the Mean  ( EMBED Equation  =  EMBED Equation  Variability of  EMBED Equation  around ( Special type of standard deviation, type of error Average amount by which  EMBED Equation  deviates from ( Less error = better, more reliable, estimate of population parameter ( EMBED Equation  influenced by two things: (1) Sample size (n) Larger n = smaller standard errors Note: when n = 1 ( ( EMBED Equation  = ( ( as starting point for ( EMBED Equation , ( EMBED Equation  gets smaller as n increases (2) Variability in population (() Larger ( = larger standard errors  Note: ( EMBED Equation  = (M   Shape of the DSM? Central Limit = DSM will approach a normal distn Theorem as n approaches infinity Very important! True even when raw scores NOT normal! True regardless of ( or ( What about sample size? (1) If raw scores ARE normal, any n will do (2) If raw scores NOT normal, n must be sufficiently large For most distributions ( n ( 30 Why are Sampling Distributions important? Tells us probability of getting  EMBED Equation , given ( & ( Distribution of a STATISTIC rather than raw scores Theoretical probability distribution Critical for inferential statistics! Allows us to estimate likelihood of making an error when generalizing from sample to popln Standard error = variability due to chance Allows us to estimate population parameters Allows us to compare differences between sample means due to chance or to experimental treatment? Sampling distribution is the most fundamental concept underlying all statistical tests Working with the Distribution of Sample Means If we assume DSM is normal If we know ( & ( We can use Normal Curve & Unit Normal Table!  z =  Example #1: ( = 80 ( = 12 What is probability of getting  EMBED Equation  ( 86 if n = 9? Example #1b: ( = 80 ( = 12 What if we change n =36 What is probability of getting  EMBED Equation  ( 86 Example #2: ( = 80 ( = 12 What  EMBED Equation  marks the point beyond which sample means are likely to occur only 5% of the time? (n = 9) Homework problems: Chapter 7: 3, 10, 11, 17     PAGE 13 PAGE 14 Figure 7-7 (p.215) The distribution of sample means for random samples of size (a) n = 1, (b) n = 4, and (c) n = 100 obtained from a normal population with = 80 and  = 20. Notice that the size of the standard error decreases as the sample size increases. Figure 7-3 (p. 205) The distribution of sample means for n = 2. 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