Question

STATS

Use the R function rnorm() to simulate selecting a random sample of size 25 from a population with mean 80 and s.d. 20. The goal here is to show how contamination affects the mean, s.d., and z-scores.

(a) Obtain the sample mean and sample sd of the simulated sample and use them to obtain the z-score for 100.

(b) Create the vector contam = c(0,seq(1000,10000,length=21)) To show the effects of contamination, separately add each value of contam to the largest value in the simulated data set, and then recompute the sample mean, sd, and z-score of 100 for each of these contaminated samples. This will result in vectors of length 22 that contain the recomputed means, sd's, and z-scores. Note that the first element of each of these vectors will be the corresponding mean, sd, and z-score of the original simulated data. For the following plots use type = "b" so that the points are connected by lines.

(c) Plot the recomputed means versus the values of contam.

(d) Plot the recomputed s.d.'s versus the values of contam.

(e) Plot the recomputed z-scores versus the values of contam.

Note: the sorted data can be obtained by

xs = sort(x,decreasing=TRUE)

You need to add each value of contam to the largest value in the sample which is xs[1]. So you will end up with 22 recomputed means, 22 recomputed s.d.'s, and 22 recomputed z-scores. An efficient way to setup this problem is to create a matrix that contains 22 columns each of which is the sorted vector xs.

nc = length(contam)

xmat = matrix(rep(xs,nc),ncol=nc)

Then add contam to the first row of that matrix to obtain the contaminated samples (the first column will be unchanged since the first element of contam is 0). Next use the apply function to obtain column means and sd's.

Answer 1

Solution v-rnorm (25, 80,20) >v-rnorm (25, 80,20) [1 60.99525 63.22160 104.05939 68.3377378.09106 72.82478 121.10129 [8] 131.97344 109.44298 121.10185 71.88081 61.31773 85.28044 98.42314 [15] 68.91232 73.58293 76.01812 82.36455 108.06920 100.27396 91.78620 22 92.74444 88.11452 90.93296 85.60073 a) m-mean (v) s-sd (v) z (100-m)/s > m-mean (v) > z-(100-m)/s 11 88.25806 [1] 19.72668 [1 0.5952317 b) contam-c (, seq (1000,10000,length-21)) > contam-c (0, seq ( 1000, 10000, length-21) ) contam 0 1000 1450 1900 2350 2800 3250 3700 4150 4600 5050 5500 [13] 5950 6400 6850 7300 7750 8200 8650 9100 9550 10000 val-list () for (i in 1:22) k-which.max (v) v [ k] [ k ] +contam [ 1 ] vec c (mean (v),sd (v), (100-mean (v))/sd(v)) val[ [il1-vec

val t111 11 88.2580575 19.7266752 0.5952317 t211 [1] 128.2580575 209.8382485 -0.1346659 t1311 11 146.2580575 299.6187870 -0.1543897 t411 [11 164.2580575 389.5006348 -0.1649755 t1511 11 182.2580575 479.4268155 -0.1715758 [1] 200.258058 569.376324 -0.176084 [11 218.2580575 659.3396131 0.1793583 tt811 [1] 236.2580575 749.3117191 -0.1818443 t1911 1 254.2580575 839.2898065 0.1837959 [101] [1] 272.2580575 929.2721379 -0.1853688 t 11]] 1 290.2580575 1019.2575893 -0.1866634 [121] [1 308.2580575 1109.2454013 -0.1877475 t[1311 1 326.2580575 1199.2350425-0.1886687 [[141] [1 344.258058 1289.226130 0.189461 15]1 1 362.2580575 1379.2183805 -0.1901498 [16]] [1 380.258058 1469.211580 -0.190754 11 398.2580575 1559.2055654 -0.1912885

t1181 1 416.2580575 1649.2002068 -0.1917645 t t1911 1 434.2580575 1739.1954028 -0.1921912 t [20] [1] 452.2580575 1829.1910715-0.1925759 470.2580575 1919.1871464 -0.1929244 1221 1 488.2580575 2009.1835730 -0.1932417 xcord- lapply (val, '[ 1) ycord contam plot (xcord,ycord,type"b")

R Graphics: Device 2 (ACTIVE) 200 300 400 100 xcord d) xcord- lapply (val, '[',2) plot (xcord, ycord, type"b)

R R Graphics: Device 2 (ACTIVE) 1000 1500 2000 xcord xcord- lapply (val, '',3) plot (xcord, ycord, type-"b")

.0 0 o-o-o-o-o-o-o-o-o.0.0.0.0.0.0.0.0.0 8 8 0009 8 000, 8

Executable R Code:

v=rnorm(25,80,20)

a)

m=mean(v)

s=sd(v)

z=(100-m)/s

b)

contam=c(0,seq(1000,10000,length=21))

v1=v

val=list()

for(i in 1:22)

{

k=which.max(v)

v[k]=v[k]+contam[i]

vec=c(mean(v),sd(v),(100-mean(v))/sd(v))

val[[i]]=vec

v=v1

}

xcord= lapply(val, '[[', 1)

ycord=contam

plot(xcord,ycord,type="b")

d)

xcord= lapply(val, '[[', 2)

plot(xcord,ycord,type="b")

e)

xcord= lapply(val, '[[', 3)

plot(xcord,ycord,type="b")