HH (version 2.1-2)

norm.curve: plot a normal or a t-curve with both x and z axes.

Description

Plot a normal curve or a t-curve with both x (with mean and se as specified) and z or t (mean=0, se=1) axes. Shade a region for rejection region, acceptance region, confidence interval. The density axis is marked in units appropriate for the z or t axis. The existence of any of the arguments se, sd, n forces dual x and (z or t) scales. When none of these arguments are used, the main title defaults to "Standard Normal Density N(0,1)" and only the z scale is printed. A second density curve, appropriate for an alternative hypothesis is displayed when the argument axis.name="z1" is specified. The shaded area is printed on the plot. When the optional argument df.t is specified, then a t-distribution with df.t degrees of freedom is plotted. norm.observed plots a vertical line with arrowhead markers at the location of the observed xbar. normal.and.t.dist is a driver function that uses all the others. In R, when RExcel is running (currently only on Windows), it has the side effect of leaving variables in the work space which are intended to be used by the normal.and.t Excel worksheet in the RExcel package for R on Windows. There are no side effects in S-Plus or in R when RExcel is not running.

Usage

normal.and.t.dist(std.dev        = 1,
                  n              = NA,
                  deg.freedom    = NA,
                  mu.H0          = 0,
                  xmin           = NA,
                  xmax           = NA,
                  fx.min         = NA,
                  fx.max         = NA,
                  Use.alpha.right= TRUE,
                  alpha.right    = .05,
                  Use.alpha.left = FALSE,
                  alpha.left     = alpha.right,
                  Use.mu.H1      = FALSE,
                  mu.H1          = NA,
                  Use.obs.mean   = FALSE,
                  obs.mean       = 0,
                  hypoth.or.conf = 'Hypoth',
                  col.mean       = 'lime green',
                  col.alpha      = 'blue',
                  col.beta       = 'red',
                  col.conf       = 'pale green',
                  col.conf.arrow = 'dark green',
                  col.mean.label = 'lime green',
                  col.alpha.label= 'blue',
                  col.beta.label = 'red',
                  col.conf.label = 'dark green',
                  cex.crit       = 1.2
                  )

norm.setup(xlim.in=c(-2.5,2.5),
           ylim.in = c(0, 0.4)/se,
           mean=0,
           main.in=main.calc,
           se=sd/sqrt(n), sd=1, n=1,
           df.t=NULL,
           Use.obs.mean=TRUE,
            ...)

norm.curve(mean=0, se=sd/sqrt(n),
          critical.values=mean + se*c(-1, 1)*z.975,
          z=if(se==0) 0 else do.call("seq", as.list(c((par()$usr[1:2]-mean)/se, length=109))),
          shade, col="blue",
          axis.name=ifelse(is.null(df.t) || df.t==Inf, "z", "t"),
          second.axis.label.line=3,
          sd=1, n=1,
          df.t=NULL,
          axis.name.expr=axis.name,
          Use.obs.mean=TRUE,
          col.label=col,
          hypoth.or.conf="Hypoth",
          col.conf.arrow=par("col"),
          col.conf.label=par("col"),
          col.crit=ifelse(hypoth.or.conf=="Hypoth", 'blue', col.conf.arrow),
          cex.crit=1.2,
          ...)

norm.observed(xbar, t.xbar, t.xbar.H1=NULL,
              col="green",
              p.val=NULL, p.val.x=par()$usr[2]+ left.margin,
              t.or.z=ifelse(is.null(deg.free) || deg.free==Inf, "z", "t"),
              t.or.z.position=par()$usr[1]-left.margin,
              cex.small=par()$cex*.7, col.label=col,
              xbar.negt=NULL, cex.large=par()$cex,
              left.margin=.15*diff(par()$usr[1:2]),
              sided="", deg.free=NULL)

norm.outline(dfunction, left, right, mu.H0, se, deg.free=NULL,
             col.mean="green")

Arguments

xlim.in, ylim.in, xmin, xmax, fx.min, fx.max

xlim, ylim. Defaults to correct values for standard Normal(0,1). User must set values for other mean and standard error.

mean

Mean of the normal distribution in xbar-scale, used in calls to dnorm.

se

standard error of the normal distribution in xbar-scale, used in calls to dnorm.

sd, std.dev, n

standard deviation and sample size of the normal distribution in x-scale. These may be used as an alternate way of specifying the standard error se.

df.t, deg.freedom

Degrees of freedom for the t distribution. When df.t is NULL, the normal distribution is used.

critical.values

Critical values in xbar-scale. A scalar value implies a one-sided test. A vector of two values implies a two-sided test.

main.in

Main title. Default value is: if (is.null(df.t)) ## normal ifelse(!(missing(se) && missing(sd) && missing(n)), paste("normal density: se =", round(se,3)), "Standard Normal Density N(0,1)") else

z

z-values (standardized to N(0,1)) used as base of plot.

shade

Valid values for shade are "right", "left", "inside", "outside", "none". Default is "right" for one-sided critical.values and "outside" for two-sided critical values.

col

color of the shaded region.

col.label, col.alpha, col.alpha.label

color of the area of the shaded rejection region and its label.

col.beta, col.beta.label

color of the area of the shaded region For Type II error and its label.

hypoth.or.conf

"Hypoth" or "Conf"

col.conf

Color of plot within confidence limits.

col.conf.arrow

Color of arrow denoting confidence limits.

col.conf.label

Color of label giving confidence level.

col.mean.label

Color of label for observed mean.

col.crit, cex.crit

Color and cex of critical values.

axis.name, axis.name.expr

defaults to "z" for the standard normal scale centered on the null hypothesis value of the mean or to "t" for the t distribution with df.t degrees of freedom. For alternative hypotheses, the user

second.axis.label.line

Defaults to 3. Normally not needed. When two curves are drawn, one normal and one t, then the second curve needs a different label for the y-axis. Set this value to 4 to avoid overprinting.

xbar, obs.mean

xbar-value of the observed data.

t.xbar

t-value of the observed data under the null hypothesis.

...

Other arguments which are ignored.

Use.obs.mean

Logical. If TRUE, then include "mean" on the plot.

alpha.right, alpha.left

Area in tail of curve.

Use.alpha.right, Use.alpha.left

Logical. If TRUE, then include the specified $\alpha$ on the plot.

t.xbar.H1

t-value under alternate hypothesis.

p.val

under specified hypothesis

p.val.x,t.or.z.position

location on x-axis to put label

t.or.z

label for axis.

cex.small

cex for left margin labels of axis.

xbar.negt

location in data scale of negative t- or z-value corresponding to observed x-value. Used for two-sided p-values.

cex.large

cex for labels in top margin.

left.margin

distance to the left of par()$usr[1].

sided

type of test.

deg.free

degrees of freedom or NULL.

dfunction

"dnorm" or "dt"

left

left end of interval

right

right end of interval

mu.H0, mu.H1

mean under the null hypothesis and alternative hypothesis.

Use.mu.H1

Logical. If TRUE, then include mu.H1 on the plot.

col.mean

Color of outline.

Examples

normal.and.t.dist()
normal.and.t.dist(xmin=-4)
normal.and.t.dist(std.dev=2)
normal.and.t.dist(std.dev=2, Use.alpha.left=TRUE, deg.free=6)
normal.and.t.dist(std.dev=2, Use.alpha.left=TRUE, deg.free=6, fx.max=.20)


old.par <- par(oma=c(4,0,2,5), mar=c(7,7,4,2)+.1)

norm.setup()
norm.curve()

norm.setup(xlim=c(75,125), mean=100, se=5)
norm.curve(100, 5, 100+5*(1.645))
norm.observed(112, (112-100)/5)
norm.outline("dnorm", 112, par()$usr[2], 100, 5)

norm.setup(xlim=c(75,125), mean=100, se=5)
norm.curve(100, 5, 100+5*(-1.645), shade="left")

norm.setup(xlim=c(75,125), mean=100, se=5)
norm.curve(mean=100, se=5, col='red')

norm.setup(xlim=c(75,125), mean=100, se=5)
norm.curve(100, 5, 100+5*c(-1.96, 1.96))

norm.setup(xlim=c(-3, 6))
norm.curve(crit=1.645, mean=1.645+1.281552, col='green',
           shade="left", axis.name="z1")
norm.curve(crit=1.645, col='red')

norm.setup(xlim=c(-6, 12), se=2)
norm.curve(crit=2*1.645, se=2, mean=2*(1.645+1.281552),
           col='green', shade="left", axis.name="z1")
norm.curve(crit=2*1.645, se=2, mean=0,
           col='red', shade="right")


par(mfrow=c(2,1))
norm.setup()
norm.curve()
mtext("norm.setup(); norm.curve()", side=1,  line=5)
norm.setup(n=1)
norm.curve(n=1)
mtext("norm.setup(n=1); norm.curve(n=1)", side=1,  line=5)
par(mfrow=c(1,1))


par(mfrow=c(2,2))

## naively scaled,
## areas under the curve are numerically the same but visually different
norm.setup(n=1)
norm.curve(n=1)
norm.observed(1.2, 1.2/(1/sqrt(1)))
norm.setup(n=2)
norm.curve(n=2)
norm.observed(1.2, 1.2/(1/sqrt(2)))
norm.setup(n=4)
norm.curve(n=4)
norm.observed(1.2, 1.2/(1/sqrt(4)))
norm.setup(n=10)
norm.curve(n=10)
norm.observed(1.2, 1.2/(1/sqrt(10)))
mtext("areas under the curve are numerically the same but visually different",
      side=3, outer=TRUE)

## scaled so all areas under the curve are numerically and visually the same
norm.setup(n=1, ylim=c(0,1.3))
norm.curve(n=1)
norm.observed(1.2, 1.2/(1/sqrt(1)))
norm.setup(n=2, ylim=c(0,1.3))
norm.curve(n=2)
norm.observed(1.2, 1.2/(1/sqrt(2)))
norm.setup(n=4, ylim=c(0,1.3))
norm.curve(n=4)
norm.observed(1.2, 1.2/(1/sqrt(4)))
norm.setup(n=10, ylim=c(0,1.3))
norm.curve(n=10)
norm.observed(1.2, 1.2/(1/sqrt(10)))
mtext("all areas under the curve are numerically and visually the same",
      side=3, outer=TRUE)

par(mfrow=c(1,1))


## t distribution
mu.H0 <- 16
se.val <- .4
df.val <- 10
crit.val <- mu.H0 - qt(.95, df.val) * se.val
mu.alt <- 15
obs.mean <- 14.8

alt.t <- (mu.alt - crit.val) / se.val
norm.setup(xlim=c(12, 19), se=se.val, df.t=df.val)
norm.curve(crit=crit.val, se=se.val, df.t=df.val, mean=mu.alt,
           col='green', shade="left", axis.name="t1")
norm.curve(crit=crit.val, se=se.val, df.t=df.val, mean=mu.H0,
           col='gray', shade="right")
norm.observed(obs.mean, (obs.mean-mu.H0)/se.val)

## normal
norm.setup(xlim=c(12, 19), se=se.val)
norm.curve(crit=crit.val, se=se.val, mean=mu.alt,
           col='green', shade="left", axis.name="z1")
norm.curve(crit=crit.val, se=se.val, mean=mu.H0,
           col='gray', shade="right")
norm.observed(obs.mean, (obs.mean-mu.H0)/se.val)



## normal and t
norm.setup(xlim=c(12, 19), se=se.val, main="t(6) and normal")
norm.curve(crit=15.5, se=se.val, mean=16.3,
           col='gray', shade="right")
norm.curve(crit=15.5, se.val, df.t=6, mean=14.7,
           col='green', shade="left", axis.name="t1", second.axis.label.line=4)
norm.curve(crit=15.5, se=se.val, mean=16.3,
           col='gray', shade="none")

norm.setup(xlim=c(12, 19), se=se.val, main="t(6) and normal")
norm.curve(crit=15.5, se=se.val, mean=15.5,
           col='gray', shade="right")
norm.curve(crit=15.5, se=se.val, df.t=6, mean=15.5,
           col='green', shade="left", axis.name="t1", second.axis.label.line=4)
norm.curve(crit=15.5, se=se.val, mean=15.5,
           col='gray', shade="none")



par(old.par)

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