12 Two Sample Hypothesis

# 12 Two Sample Hypothesis - Two-Sample Hypothesis Equal and...

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Unformatted text preview: Two-Sample Hypothesis Equal and Unequal Variances (D&B Chp. 10) Difference Between Two Means Data: trout1.txt, trout2.txt & lead.washington.txt EqualScript: Lec12.TwoSampEqV.ssc UnEqualScript: Lec12.TwoSampUnEqV.ssc Brown Trout ( Salmo trutta ) Trout Length at Stocking ¡ 1977 Population : trout1 ¢ n = 99 ¢ mean = 136.5 mm ¢ var = 218.9 mm 2 ¡ 1978 Population : trout2 ¢ n = 100 ¢ mean = 207.8 mm ¢ var = 273.8 mm 2 Is there a difference? ¡ Concern that size at stocking will affect survivorship. ¡ First we must see if, in fact, they are different in size. 100 150 250 5 10 15 100 150 200 250 0 2 4 6 8 1 2 Length (mm) Trout 1 Trout 2 Histogram Script my.breaks = seq(95,250,by=5) par(mfrow=c(2,1)) par(mar=c(0,4,4,2)) hist(trout1,breaks=my.breaks,cex=1.3) box() par(mar=c(5,4,0,2)) hist(trout2,breaks=my.breaks, xlab="Length (mm)",cex=1.3) box() par(mar=c(5,4,4,2)+.1) 100 150 200 250 1977 1978 Length (mm) n = 99 n = 100 Boxplot Script boxplot(trout1,trout2, names=c("1977","1978"), ylab="Length (mm)") text(45,mean(trout1), paste("n =",length(trout1))) text(90,mean(trout2), paste("n =",length(trout2))) Other Displays of Information plot(c(0,3),c(0,6), type='n',xlab="Examples",ylab="Length") boxes(x=c(1),width=c(0.25), stats=cbind(c(1,2,3,4,5))) points(2,3,pch=15,col=1) lines(c(2,2),c(2,4),lwd=3) Examples Length 0.0 0.5 1.0 1.5 2.0 2.5 3.0 1 2 3 4 5 6 Other Statistics range(trout1) 99.06 175.26 range(trout2) 160.02 246.38 stdev(trout1) 14.79436 stdev(trout2) 16.54675 Year Length (mm) 100 150 200 250 1977 1978 CI Boxes Script plot(c(0,3),c(100,250), axes=F,type='n',ylab="Length (mm)",xlab="Year") axis(side=2) axis(side=1,at=c(1,2),labels=c("1977","1978")) box() boxes(c(1,2),width=0.25, cbind( c(range(trout1)[1], mean(trout1)-2*stdev(trout1)/sqrt(length(trout1)), mean(trout1), mean(trout1)+2*stdev(trout1)/sqrt(length(trout1)), range(trout1)[2]), c(range(trout2)[1], mean(trout2)-2*stdev(trout2)/sqrt(length(trout2)), mean(trout2), mean(trout2)+2*stdev(trout2)/sqrt(length(trout2)), range(trout2)[2]))) 100 150 200 250 1977 1978 Length (mm) Boxplot with Notch boxplot(trout1,trout2,notch=T, names=c("1977","1978"),ylab="Length (mm)") But is there a difference? 2 1 2 1 : : μ μ μ μ ≠ = A H H Test Statistic 2 1 ) ( 2 1 2 1 x x SE x x z − − − − = μ μ Difference? 2 1 x x − ? ) ( 2 1 = − x x V The difference represents a statistic, that under the null hypothesis of no difference should have a mean equal to zero. Variance of sum is sum of the variances. Assumptions to Check ¡ If the samples came from populations having normal distributions ¡ If the two populations have equal variances Then 2 1 2 1 2 2 2 1 2 2 1 2 1 2 1 υ υ + + = + = − = − − SS SS s n s n s s where s x x t p p p x x x x Are they normal populations?...
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## This note was uploaded on 11/12/2009 for the course BTRY 3010 at Cornell.

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12 Two Sample Hypothesis - Two-Sample Hypothesis Equal and...

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