Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Foundations of statistics wikipedia , lookup
History of statistics wikipedia , lookup
Taylor's law wikipedia , lookup
Degrees of freedom (statistics) wikipedia , lookup
Bootstrapping (statistics) wikipedia , lookup
Law of large numbers wikipedia , lookup
Central limit theorem wikipedia , lookup
Resampling (statistics) wikipedia , lookup
Statistics The difference of two means April 27, 2009 1. Assumptions: X1 , . . . , Xm and Y1 , . . . , Yn are independent random samples from populations that have a normal distribution with unknown means µX , µY and unknown variances. 2. The key fact is this: under these assumptions, X − Y − (µX − µY ) q 2 ∼ Norm(0, 1) . 2 σX σY + m n This follows from the fact that variances and means add and that the sum of independent normal random variables is normal. 3. We replace σX and σY by the corresponding sample standard deviations to get that this random variable X − Y − (µX − µY ) q 2 2 SX SY m + n has approximately a t-distribution with ν degrees of freedom where ν is ν= 2 SX m 2 /m)2 (SX m−1 2 SY n + + 2 2 /n)2 (SY n−1 4. Insert long story here about “old-fashioned” practice and the Behrens-Fisher problem. 5. Confidence intervals for µ1 − µ2 : r ∗ x−y±t s2 s2X + Y m n ! t∗ = tα/2,ν 6. Robustness. Homework - due Thursday, April 30, 2009 1. Read Section 7.2. 2. Do problems 7.4,5,6. > s=getdata(’subliminal’) > s[c(1,2),] group before after 1 T 18 24 2 T 18 25 > t.test(after-before~group,data=s) Welch Two Sample t-test data: after - before by group t = -1.9136, df = 13.919, p-value = 0.07647 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -6.6825702 0.3825702 sample estimates: mean in group C mean in group T 8.25 11.40