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Sampling Error think of sampling as benefiting from room for movement Sampling Error think of sampling as benefiting from room for movement Population µ Sample _ X _ The population mean is µ. The sample mean is X. Population µ Sample _ X s _ The population mean is µ. The sample mean is X. The population standard deviation is , the sample sd is s. Sample _ C XC Sample _ D XD Population Sample _ B µ Sample _ E XE XB Sample _ A XA In reality, the sample mean is just one of many possible sample means drawn from the population, and is rarely equal to µ. Sample _ C XC sc Sample _ D XD sd Population Sample _ B µ Sample _ E XE se XB sb Sample _ A XA sa In reality, the sample mean is just one of many possible sample means drawn from the population, and is rarely equal to µ. Sampling error = Statistic - Parameter _ Sampling error for the mean = X - µ Sampling error for the standard deviation = s - Unbiased and Biased Estimates An unbiased estimate is one for which the mean sampling error is 0. An unbiased statistic tends to be neither larger nor smaller, on the average, than the parameter it estimates. _ The mean X is an unbiased estimate of µ. The estimates for the variance s2 and standard deviation s have denominators of N-1 (rather than N) in order to be unbiased. 2 SS = N s2 SS = (N - 1) s SS = (N - 1) Correlations describe co-variation on two variables. What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 SCI 40 20 80 60 CIV 40 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 SCI 40 20 80 60 CIV 40 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 .69 60 MATH 40 80 60 SCI 40 20 80 60 CIV 40 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 SCI 40 20 80 60 CIV 40 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 .68 SCI 40 20 80 60 CIV 40 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 SCI 40 20 80 60 CIV 40 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 SCI 40 20 80 60 .60 40 CIV 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 SCI 40 20 80 60 CIV 40 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 SCI 40 20 80 60 CIV 40 20 2 .13 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 SCI 40 20 80 60 CIV 40 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 .65 SCI 40 20 80 60 CIV 40 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 SCI 40 20 80 60 CIV 40 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 SCI 40 20 80 60 .51 40 CIV 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 SCI 40 20 80 60 CIV 40 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 SCI 40 20 80 60 CIV 40 20 2 .16 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 SCI 40 20 80 60 CIV 40 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 SCI 40 20 80 60 .51 40 CIV 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 SCI 40 20 80 60 CIV 40 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 SCI 40 20 80 60 CIV 40 20 2 .15 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 SCI 40 20 80 60 CIV 40 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 What’s the correlation? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 60 MATH 40 80 60 SCI 40 20 80 60 CIV 40 20 2 .06 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 Rounded, the mean reading score is 52 and sd=10 and the mean math score is also 52 and sd=10. If a given student’s reading score is 67, then what is your best estimate of her math score? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 .69 60 MATH 40 80 60 SCI 40 20 80 60 CIV 40 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 In predicting someone’s math score, if you could have just one piece of information, and it is either (a) her reading score, or (b) her self concept score, which would you rather have? What’s it worth to you? 40 60 80 20 40 60 80 80 60 RDG 40 20 80 .69 60 .16 MATH 40 80 60 SCI 40 20 80 60 CIV 40 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 Explain to a non-statistician what it means to say “reading and math scores are correlated r=.69 in this population”. 40 60 80 20 40 60 80 80 60 RDG 40 20 80 .69 60 MATH 40 80 60 SCI 40 20 80 60 CIV 40 20 2 0 CONCPT -2 20 40 60 80 20 40 60 80 -2 0 2 A key to understanding r zy = zx When X and Y are perfectly correlated We can say that zx perfectly predicts zy zy’ = zx Or ^ zy = zx When they are imperfectly correlated, i.e., rxy ≠ 1 or -1 zy’ = rxyzx r is the slope of the predicted line, with a zero-intercept of z’y=0 Explain to a non-statistician what it means to say “reading and math scores are correlated r=.69 in this population”.