on Inference the Means of 2 Populations, Variances Known Assumptions: Both samples are random samples, independent, and are normal (if not normal, C.L.T. apply) E(Xbar1 Xbar2) = E(Xbar1) E(Xbar2) = 1-2 V(Xbar1-Xbar2)=V(Xbar1)+V(Xbar2)= Alt. Hyp: != delta_0: P-value: Sum of P above t_0 and below t_0. Reject if t_0>ta/2,n1+n2-2 or t_0<ta/2,n1+n2-2 Dif-means > delta_0: P-value is P above t_0, reject if t_0 > ta,n1+n2-2 Dif-means < delta_0: P-value is P below t_0, reject if t_0 < -ta,n1+n2-2 Test Statistic for Difference in Means of Two Normal Distributions, Variances Unknown, not necessarily Equal
Hypothesis Testing on the Difference in Means, Variances Known Null hypothesis: 1-2 = delta_0 Test stat:
Alternative Hyp: != delta_0P-value: P above Z_0 and P below Z_0, P = 2[1-phi(|Z_0|)] Rejection: Z_0 > Z_a/2 or Z_0 < -Z_a/2 Hyp > Delta_0P-value: P above Z_0, P = 1phi(Z_0), reject Z_0 > Z_a Hyp < Delta_0P-value: P below Z_0, P = phi(Z_0), reject Z_0 < Z_a Type II Error and Choice of Sample Size Sample Size for 2-sided Alt Hyp. On Difference in Means, Variances known, n1=n2
Degrees of freedom:
Confidence Interval on Difference in Means Case 1: CI on DiM of2 Norm Distns, Variances Unknown and Equal
For one-sided:
Where Sp= Case 2: CI on DiM of 2 Norm Distns, Variances Unknown and Unequal
Confidence Intervals on the Difference in Means, Variances Known
V as defined above For one-sided bounds, replace /2 with just for proper bound. Lower confidence = -t, upper = t. Paired t-Test - When observations on two populations are collected in pairs D=E(X1-X2) = E(X1)-E(X2)=1-2 Null hyp: D=delta_0 Test stat:
Sample Size for a specified E on the Difference in Means, and variances known when n1=n2
Alt. hyp: != delta_0, P-value is sum of P above t_0 and below t_0, reject for t_0 > ta/2,n-1 or t_0 < -ta
/2,n-1
Inference on the Means of Two Populations, Variances Unknown Hypothesis Testing on the Difference in Means Pooled estimator of 2 = Sp2:
Hyp > delta_0, p-value is P above t_0, reject if t_0 > ta,n-1 Hyp < delta_0, p-value is P below t_0, reject if t_0 < -ta,n-1 D-bar is sample average of the n differences D1, D2,Dn. SD is sample standard dev. Of the diff. CI on D for paired observations
Inferences the on Ratio of Variances of Two Normal Populations F distribution: where W and Y are independent chi-square random vars with u and v DoF. F has pdf:
SST=SSTreatments+SSE where SST=total SS (above), SStreatments=nSUM (from above simplified eqn) and SSE=SUMSUM from above simplified eqn. Testing Hypotheses on more than 2 Means (ANOVA) MSTreatments=SSTreatments/(a-1), E(MSTreatments)= MSE=SSE/a(n-1), E(MSE)=variance (sigmasquared) H0: tau1=tau2==taun=0, H1: taui!= 0 for at least one i Test stat: F0=MSTreatments/MSE P-value: Probability beyond f0 in the Fa-1,a(n-1)distn. (not alpha) Rejection criterion: f0>falpha,a-1,a(n-1) Computing Formulae: SST= , SSTreatments= , SSE=SSTSSTreatments For unequal sample sizes: SST= , SSTreatments= , SSE=SSTSSTreatments Randomized complete block design
Fu,v, u in numerator, v in denom. Mean for F distribution: =v/(v-2) for v > 2 and
p. 245!..f1-a,u,v=1/fa,v,u Test stat.:
Testing Hypotheses on the Equality of Variances of Two Normal Distributions Null hyp: 12=22, test stat Alt. hyp. 12!=22, reject if 12>22, reject if 12<22, reject if C.I. on the Interval of the Ratio of Variances of 2 Normal Distributions
where SST=SSTreatments+SSBlocks+SSE (just as above) DoF for above follows: ab-1=(a-1)+(b-1)+(a-1)(b1) MSTreatments=SSTreatments/(a-1), MSBlocks=SSBlocks/(b-1), MSE=SSE/(a-1)(b-1) E(MSTreatments)= , E(MSBlocks)= , E(MSE)=sigma-squared Computing Formulae: SST= , SSTreatments= , SSBlocks= , SSE=SST-SSTreatments-SSBlocks
Hyp. Testing on the Equality of 2 Binomial Proportions
P-hat1 = X1/n1 and P-hat2 =X2/n2. For testing p1=p2 If p1=p2 (H0 is true), then p1-p2 cancels, etc, and p-hat = (X1+X2)/(n1+n2) Rejection: double-sided: |Z0| > Za/2, p1>p2, Z0>Za [rev for <] P-val: double: P=2[1-phi(|Z0|)], p1>p2: P=1phi(Z0), p1<p2: P=phi(Z0) Type 2 Error for 2 Bin Prop: NEEDS FOR MIDTERM? CI on Difference in Bin. Prop:
Residuals: eij=yij-y-hatij, fitted values: y-hatij=ybari.+y-bar.j-y-bar.. TREATMENTS ARE LABELED AS 1 TO a. BLOCKS ARE LABELED AS 1 TO b. ROWS GO FROM LEFT-TO-RIGHT, TOP-TO-BOTTOM AS i to n
ANOVA Going left to right, top to bottom: yij. Sum of row: yi., average of row: . Totals: y.. for sum, y-bar.. for average Sum of Squares identity: