1. Simulate data for 30 draws from a normal distribution where the means and standard deviations vary among three distributions

set.seed(1)
a=rnorm(30,mean=c(0,1,10),sd=c(1,10,100))
a
##  [1]   -0.62645381    2.83643324  -73.56286124    1.59528080    4.29507772
##  [6]  -72.04683841    0.48742905    8.38324705   67.57813517   -0.30538839
## [11]   16.11781168   48.98432364   -0.62124058  -21.14699887  122.49309181
## [16]   -0.04493361    0.83809737  104.38362107    0.82122120    6.93901321
## [21]  101.89773716    0.78213630    1.74564983 -188.93516959    0.61982575
## [26]    0.43871260   -5.57955067   -1.47075238   -3.78150055   51.79415602

2. Simulate 2 continuous variables (normal distribution) (n=20) and plot the relationship between them

set.seed(2)
b=rnorm(20,0,1)
set.seed(3)
c=rnorm(20,0,1)
plot(c~b)

3. Simulate 3 variables (x1, x2 and y). x1 and x2 should be drawn from a uniform distribution and y should be drawn from a normal distribution. Fit a multiple linear regression.

set.seed(4)
x1=runif(100,1,10)
x1
##   [1] 6.272203 1.080512 3.643657 3.496375 8.322168 3.343850 7.519653
##   [8] 9.154829 9.541362 1.658300 7.792075 3.574006 1.900482 9.586619
##  [15] 4.740464 5.095922 9.739501 6.255892 9.659842 7.855322 7.430577
##  [22] 9.969516 5.556438 5.409489 6.842452 8.477258 5.337991 8.575716
##  [29] 5.623316 5.767998 6.104010 3.150540 8.901963 6.890698 5.341338
##  [36] 9.739268 5.140233 6.598508 4.495762 1.059335 9.444872 3.179540
##  [43] 6.090351 2.628092 9.140494 1.758622 9.110545 9.024503 7.510374
##  [50] 6.083289 4.497490 7.713360 9.058789 8.281691 8.366917 4.790800
##  [57] 2.590268 2.564079 9.032912 7.690007 6.048855 1.641104 8.682263
##  [64] 9.223399 3.028251 6.657041 1.622305 5.633686 8.248756 9.746105
##  [71] 4.086237 6.682271 4.690488 4.111247 8.423856 7.209105 3.885805
##  [78] 4.971967 3.356969 2.198840 9.194308 7.382511 6.148237 9.252714
##  [85] 9.117191 1.527598 1.411309 9.899277 2.852425 9.361504 2.748732
##  [92] 2.141208 5.854319 3.451034 4.182280 1.756944 7.888939 5.000374
##  [99] 1.325237 7.311602
x2=runif(100,100,200)
x2
##   [1] 125.3427 162.9631 126.6404 153.2446 146.8265 157.4397 166.8445
##   [8] 121.2114 197.5632 180.9383 127.5334 139.9124 129.0300 149.2993
##  [15] 175.6775 151.0697 143.8149 179.0914 191.1315 107.1074 114.2642
##  [22] 139.3073 185.6438 126.4254 109.4638 172.4334 198.0477 120.0600
##  [29] 155.2264 134.5464 140.8390 118.8031 134.5553 118.3656 181.3115
##  [36] 188.1635 170.0615 127.6427 143.2004 101.2816 156.3038 167.0700
##  [43] 131.3602 194.7544 116.8803 144.0414 157.1627 197.4654 176.4771
##  [50] 144.3055 135.5862 103.6457 159.3880 196.1553 125.2730 184.7206
##  [57] 121.2781 127.7891 147.9385 178.0000 190.0940 194.3461 141.5215
##  [64] 133.4110 128.2732 107.2222 107.0682 115.8950 115.0863 175.2092
##  [71] 109.5687 161.1492 120.0944 150.3318 112.9100 100.2923 164.3842
##  [78] 124.1036 142.0035 171.5428 110.0685 160.6837 121.2432 189.4741
##  [85] 156.3198 124.7725 173.0656 126.6332 175.4257 195.7943 148.1236
##  [92] 190.9495 199.0105 123.2417 128.1779 170.3925 183.3597 146.5679
##  [99] 139.0684 109.3888
y=rnorm(100,0,1)
y
##   [1]  0.6848019360 -0.1151135095 -0.3564751798 -0.1057716076  0.0448827901
##   [6] -1.7261732320  1.5557870203  0.7764126917 -1.0985075088 -1.7280197536
##  [11]  0.4276382246  0.7445646452  0.8652207970  0.3053288101 -0.1140227912
##  [16]  0.4236522402 -0.7977096868 -0.6041972494  1.7150105938 -0.7159482778
##  [21] -0.1332356122 -0.9997650626  1.8737601171 -0.3373884320  0.9732702887
##  [26]  0.9878279309 -0.9412566085  0.3491855939 -0.5944186817 -2.3822428313
##  [31]  1.0780189737  0.6682451050 -0.9646256667 -1.9752373319 -0.5847739007
##  [36]  0.9692770362  0.5522923259 -0.0821555007 -1.6767137584  1.2126074270
##  [41]  1.0004998710  0.7193289908 -0.8443641520  0.6219853903 -0.7226137804
##  [46] -0.4494786251 -1.1955060501  0.3904723630 -0.5163766426  0.9098689779
##  [51]  0.8769846530 -0.8161958099  1.5392932699  1.3745257156 -0.4832487112
##  [56]  0.5503499503 -0.8573656630 -0.7069613662 -2.0970775334  1.0994367548
##  [61]  0.3420340890  0.4908294804 -0.9319990260 -1.4278919839  0.9757650946
##  [66] -1.5463411878  0.0177034792 -0.7747174012 -0.2293422872 -0.2743821044
##  [71]  1.7960637815 -0.4781128994 -0.5947628530 -2.2579382170  1.6826072118
##  [76]  0.0722906844 -0.4400240932  0.6265733926 -0.7997960594 -1.1279860222
##  [81] -1.0250160534  0.0710717295  0.3817111616 -1.6225883175  1.9005426699
##  [86] -0.7161791664  0.3804596689  0.4408428474  0.2573258583 -0.1794485371
##  [91] -0.6901276793 -0.0004228025  0.5655808964 -1.2087470098 -0.3461711560
##  [96] -0.6501970444 -0.8895916708  1.4770298873 -1.1954751385  1.7504948348
mlr=lm(y~x1+x2)
mlr
## 
## Call:
## lm(formula = y ~ x1 + x2)
## 
## Coefficients:
## (Intercept)           x1           x2  
##   -0.597013     0.023268     0.002585
plot(mlr)

#4. Simulate 3 letters repeating each letter twice, 2 times.

rep(letters[1:3],each=2,times=2)
##  [1] "a" "a" "b" "b" "c" "c" "a" "a" "b" "b" "c" "c"

5. Create a dataframe (n = 27) with 3 groups, 2 factors and two quantitative response variables. Use the replicate function.

datfr=data.frame(group=rep(letters[1:3]),factor=rep(letters[4:5]), x=rnorm(6,0,1),y=rnorm(6,1,2))
datfr
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753
replic=replicate(10, expr = datfr, simplify = FALSE)
replic
## [[1]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753
## 
## [[2]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753
## 
## [[3]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753
## 
## [[4]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753
## 
## [[5]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753
## 
## [[6]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753
## 
## [[7]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753
## 
## [[8]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753
## 
## [[9]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753
## 
## [[10]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753

replicate

replic=replicate(10, expr=datfr, simplify=FALSE)
replic
## [[1]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753
## 
## [[2]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753
## 
## [[3]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753
## 
## [[4]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753
## 
## [[5]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753
## 
## [[6]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753
## 
## [[7]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753
## 
## [[8]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753
## 
## [[9]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753
## 
## [[10]]
##   group factor          x          y
## 1     a      d  1.2147301 -3.8441746
## 2     b      e -1.5478003  2.1126566
## 3     c      d -0.3022460  3.2110678
## 4     a      e  1.0392077  1.3328736
## 5     b      d -0.7678417  0.5490754
## 6     c      e  1.5246726  0.5431753