((using R progrem do the following))

5. F distribution. Compute the following probabilities and quantiles for the F8,5 distribution.

(a) P(F8,5 < 3). 3.4. EXERCISES 91

(b) P(F8,5 > 4).

(c) P(1 < F8,5 < 6).

(d) The quantiles f0.025, f0.5, and f0.975.

((script ))

# 3.2.5 F-Distribution ########################################

# Example 1. When the two population variances are in fact equal... var(golub[1042,golubFactor=="ALL"])/var(golub[1042,golubFactor=="AML"])

pf(0.7116441, 26, 10) # left-side tail of the F-distribution cumulative density function (CDF)

f<-function(x) { df(x,26,10) } plot(f, # function 0, # start 10, # end cex.lab=1.5, # make axis labels big xlab="x", ylab="F-Distribution probability density function f(x)") mtext("0.71",side=1,at=.7,cex=1, col="red") x1 <- seq(0,0.71,0.01) x2 <- seq(0.71,10,0.01) polygon(c(0,x1,.71), c(0,f(x1),0), col="magenta") polygon(c(.71,x2,10), c(0,f(x2),0), col="lightblue") arrows(4.5,.50,0.55,.3, lwd=3, col="green") text(5.0, 0.53 - c(0,.11), cex = 1.2, adj=c(0,0), col="blue", c(expression(P(X <= 0.71)), expression(paste(" = ", integral(f(x) * dx, 0, 0.71)))))

f<-function(x) { pf(x,26,10) } plot(f, # function 0, # start 10, # end cex.lab=1.5, # make axis labels big col="red", lwd=6, # make line thicker xlab="x", ylab="F-Distribution cumulative distribution function F(x)") mtext("0.71",side=1,at=.7, cex=1, col="red") mtext("0.23",side=2,at=.23,cex=1, col="red") arrows(3,0.3,0.71,0.23, lwd=3, col="green") text(4.0, 0.5 - c(0,.12,.24,.36,.48), cex=1.2, adj=c(0,0), col="blue", c(expression(P(X <= 0.71)), expression(paste(" = ", integral(f(x) * dx, 0, 0.71))), expression(paste(" = ", bgroup("[", F(x) ,"]")[0]^0.71)), expression(paste(" = ", CDF(0.71))), expression(paste(" = ", 0.23))))

qf(0.025, 26, 10)