Installing packages

library(ggplot2)
library(dplyr)
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
library(readxl)

Setting Directory

getwd()
## [1] "/Users/keerthichereddy/Documents/Trim1/ASM/WineProject"
setwd("/Users/keerthichereddy/Documents/Trim1/ASM/WineProject/")
red_wine <- read_excel("/Users/keerthichereddy/Documents/Trim1/ASM/WineProject/winequality-red.xls")
dim(red_wine)
## [1] 1599   12
sum(is.na("red_wine"))
## [1] 0

1. Suppose the population mean of the variable “density” is μ , do the following inferences:

a. Provide an estimate of μ based on the sample;

mu.density <- mean(red_wine$density)
mu.density
## [1] 0.9967467

b. Use the Central Limit Theorem (CLT) to quantify the variability of your estimate;

sd.density <- sd(red_wine$density)
sd.density
## [1] 0.001887334
var.density <- sd(red_wine$density)/sqrt(length(red_wine$density))
var.density
## [1] 4.71981e-05

c. Use the CLT to give a 95% confidence interval for μ.

hist(red_wine$density)

#conf_quantile <- quantile(mu.density, probs=c(0.025, 0.975))
#conf_quantile
#why are the values different from lower and upper

lc.denisty <- mu.density - 2*(var.density)
lc.denisty
## [1] 0.9966523
uc.density <- mu.density + 2*(var.density)
uc.density
## [1] 0.9968411

d. Use the bootstrap method to do parts b and c, and compare the results

mu.density.set <- NULL
for (k in 1:2000) {
  density.bootstrap <- sample(red_wine$density, size=1599, replace=T)
  mu.density <- mean(density.bootstrap)
  mu.density.set[k] <- mu.density
}

sd(mu.density.set)
## [1] 4.813814e-05
var.density.set <- var(mu.density.set)
var.density.set
## [1] 2.31728e-09
conf.q.denisty <- quantile(mu.density.set, probs=c(0.025, 0.975))
conf.q.denisty
##      2.5%     97.5% 
## 0.9966464 0.9968388
lower <- mu.density.set - 2*(var.density.set)
lower
##    [1] 0.9967614 0.9967688 0.9967200 0.9967183 0.9966774 0.9968033 0.9967663
##    [8] 0.9969063 0.9967109 0.9967694 0.9967769 0.9967441 0.9966910 0.9966458
##   [15] 0.9967292 0.9967318 0.9968028 0.9967832 0.9967664 0.9967837 0.9967461
##   [22] 0.9967228 0.9968042 0.9968051 0.9967747 0.9967699 0.9967297 0.9967589
##   [29] 0.9967398 0.9967728 0.9967669 0.9967297 0.9968005 0.9966556 0.9967871
##   [36] 0.9967340 0.9967553 0.9967121 0.9966997 0.9967970 0.9967840 0.9966496
##   [43] 0.9967391 0.9966814 0.9967969 0.9967209 0.9967375 0.9967936 0.9966770
##   [50] 0.9966683 0.9967974 0.9967772 0.9967315 0.9967526 0.9967868 0.9967735
##   [57] 0.9967705 0.9967185 0.9967266 0.9967440 0.9967231 0.9967660 0.9966411
##   [64] 0.9967141 0.9968102 0.9968279 0.9967154 0.9966315 0.9967939 0.9967784
##   [71] 0.9968183 0.9967855 0.9967917 0.9967928 0.9967744 0.9968043 0.9967147
##   [78] 0.9967671 0.9967319 0.9966838 0.9967308 0.9966792 0.9967779 0.9967641
##   [85] 0.9966816 0.9967438 0.9968048 0.9967552 0.9967580 0.9967986 0.9967210
##   [92] 0.9967533 0.9967243 0.9967522 0.9967296 0.9967594 0.9968117 0.9967937
##   [99] 0.9967663 0.9967764 0.9966496 0.9966964 0.9968137 0.9966461 0.9966585
##  [106] 0.9967136 0.9966859 0.9967904 0.9968072 0.9967050 0.9967695 0.9967939
##  [113] 0.9967145 0.9966570 0.9966732 0.9967785 0.9966871 0.9967497 0.9967482
##  [120] 0.9967467 0.9966893 0.9966966 0.9967329 0.9967671 0.9967034 0.9967243
##  [127] 0.9968035 0.9967500 0.9967266 0.9966547 0.9967175 0.9967156 0.9967034
##  [134] 0.9967792 0.9967876 0.9967080 0.9967640 0.9967867 0.9966659 0.9967986
##  [141] 0.9967306 0.9967365 0.9967545 0.9967496 0.9968205 0.9966970 0.9967762
##  [148] 0.9967592 0.9966969 0.9966759 0.9967403 0.9967115 0.9967716 0.9967314
##  [155] 0.9967551 0.9966559 0.9968233 0.9967621 0.9967279 0.9966588 0.9966766
##  [162] 0.9967685 0.9966973 0.9967802 0.9967727 0.9967752 0.9966979 0.9967077
##  [169] 0.9967361 0.9967735 0.9967576 0.9967394 0.9967322 0.9966803 0.9966296
##  [176] 0.9967764 0.9967856 0.9967560 0.9967031 0.9967819 0.9968294 0.9966561
##  [183] 0.9967771 0.9967328 0.9968088 0.9967011 0.9967178 0.9967469 0.9966738
##  [190] 0.9967674 0.9967272 0.9967630 0.9967474 0.9967917 0.9967416 0.9968388
##  [197] 0.9966857 0.9967679 0.9967394 0.9967022 0.9967689 0.9967933 0.9966535
##  [204] 0.9968051 0.9968337 0.9967627 0.9968279 0.9967631 0.9967390 0.9966332
##  [211] 0.9966651 0.9966513 0.9967207 0.9967712 0.9967654 0.9967832 0.9967940
##  [218] 0.9967318 0.9967377 0.9967062 0.9967155 0.9967696 0.9966742 0.9967980
##  [225] 0.9967477 0.9967851 0.9968713 0.9967464 0.9967163 0.9967415 0.9966661
##  [232] 0.9967987 0.9967436 0.9967775 0.9966972 0.9967862 0.9967649 0.9967987
##  [239] 0.9967509 0.9967483 0.9967587 0.9967644 0.9967748 0.9967275 0.9967251
##  [246] 0.9967291 0.9967095 0.9967703 0.9967573 0.9967500 0.9967472 0.9967465
##  [253] 0.9968509 0.9968158 0.9967352 0.9967784 0.9967562 0.9967460 0.9968394
##  [260] 0.9967138 0.9967533 0.9967990 0.9967152 0.9967905 0.9967173 0.9966944
##  [267] 0.9967650 0.9967566 0.9967456 0.9967157 0.9966441 0.9967265 0.9967921
##  [274] 0.9966791 0.9967655 0.9967104 0.9967020 0.9967846 0.9967204 0.9968128
##  [281] 0.9967488 0.9967594 0.9967108 0.9967639 0.9967387 0.9967932 0.9967271
##  [288] 0.9968367 0.9966752 0.9967420 0.9966847 0.9967851 0.9966877 0.9968178
##  [295] 0.9968031 0.9966464 0.9967853 0.9967156 0.9967545 0.9967902 0.9966878
##  [302] 0.9967632 0.9966850 0.9967913 0.9967881 0.9967567 0.9967303 0.9967814
##  [309] 0.9967302 0.9966989 0.9967194 0.9967951 0.9967338 0.9968033 0.9968194
##  [316] 0.9967114 0.9967791 0.9966916 0.9967353 0.9968214 0.9967433 0.9967489
##  [323] 0.9968361 0.9967651 0.9966776 0.9966330 0.9967216 0.9967768 0.9967225
##  [330] 0.9967127 0.9967218 0.9967820 0.9967473 0.9967302 0.9967516 0.9967801
##  [337] 0.9967849 0.9967650 0.9967633 0.9967254 0.9967237 0.9967788 0.9967162
##  [344] 0.9968131 0.9967368 0.9967929 0.9967866 0.9967126 0.9968523 0.9966773
##  [351] 0.9967075 0.9967252 0.9967205 0.9968054 0.9967705 0.9967634 0.9967317
##  [358] 0.9967487 0.9967899 0.9968047 0.9967498 0.9967803 0.9966997 0.9968256
##  [365] 0.9968065 0.9967868 0.9967002 0.9966894 0.9968003 0.9966602 0.9967082
##  [372] 0.9968193 0.9967190 0.9967611 0.9967366 0.9966981 0.9967060 0.9966292
##  [379] 0.9966920 0.9967886 0.9967357 0.9967491 0.9967231 0.9967737 0.9967794
##  [386] 0.9967725 0.9967731 0.9967050 0.9967213 0.9967826 0.9967166 0.9966915
##  [393] 0.9967048 0.9967541 0.9966966 0.9967567 0.9967401 0.9967721 0.9967127
##  [400] 0.9967431 0.9968029 0.9967538 0.9968191 0.9968106 0.9967462 0.9967429
##  [407] 0.9967199 0.9967024 0.9968097 0.9966804 0.9968758 0.9968366 0.9966906
##  [414] 0.9967503 0.9966912 0.9968675 0.9967194 0.9967729 0.9966922 0.9966241
##  [421] 0.9968177 0.9966528 0.9967557 0.9967218 0.9966910 0.9967433 0.9967688
##  [428] 0.9968247 0.9967140 0.9968269 0.9967063 0.9968519 0.9967086 0.9967210
##  [435] 0.9967568 0.9967444 0.9966496 0.9967161 0.9967297 0.9966547 0.9967421
##  [442] 0.9967274 0.9966922 0.9968385 0.9967343 0.9967355 0.9967361 0.9968839
##  [449] 0.9967760 0.9967057 0.9967617 0.9967095 0.9967367 0.9967147 0.9968040
##  [456] 0.9967551 0.9967393 0.9967179 0.9967131 0.9967376 0.9967590 0.9967423
##  [463] 0.9967112 0.9967122 0.9967321 0.9967810 0.9967767 0.9968999 0.9967711
##  [470] 0.9967397 0.9967644 0.9968147 0.9967249 0.9966674 0.9967102 0.9966737
##  [477] 0.9968181 0.9967059 0.9967883 0.9968140 0.9967180 0.9967907 0.9968007
##  [484] 0.9967759 0.9968112 0.9967554 0.9967670 0.9967328 0.9968031 0.9967081
##  [491] 0.9966770 0.9967791 0.9967253 0.9967532 0.9967796 0.9967340 0.9967078
##  [498] 0.9967325 0.9967624 0.9966741 0.9967272 0.9967594 0.9967719 0.9967724
##  [505] 0.9967327 0.9968090 0.9967543 0.9967091 0.9967086 0.9967574 0.9966818
##  [512] 0.9968103 0.9967751 0.9967436 0.9967119 0.9966114 0.9967485 0.9967346
##  [519] 0.9967544 0.9967356 0.9967942 0.9967666 0.9967556 0.9967458 0.9967195
##  [526] 0.9967942 0.9966714 0.9966841 0.9967162 0.9967367 0.9966805 0.9967336
##  [533] 0.9968055 0.9967622 0.9967301 0.9967418 0.9966896 0.9966351 0.9967578
##  [540] 0.9967603 0.9967121 0.9968117 0.9967280 0.9967775 0.9967313 0.9967283
##  [547] 0.9967079 0.9968100 0.9967606 0.9967211 0.9967487 0.9967264 0.9966914
##  [554] 0.9968036 0.9967193 0.9967046 0.9967370 0.9968220 0.9968761 0.9967370
##  [561] 0.9967513 0.9966937 0.9967315 0.9967447 0.9967409 0.9966787 0.9967257
##  [568] 0.9967252 0.9967367 0.9967718 0.9968455 0.9967155 0.9967487 0.9967493
##  [575] 0.9967004 0.9967129 0.9967589 0.9968185 0.9966847 0.9967466 0.9967491
##  [582] 0.9967393 0.9967203 0.9967195 0.9968253 0.9966861 0.9968455 0.9967509
##  [589] 0.9967470 0.9966972 0.9967634 0.9967199 0.9967259 0.9966880 0.9967602
##  [596] 0.9967662 0.9966214 0.9967244 0.9967448 0.9967649 0.9967818 0.9967776
##  [603] 0.9967262 0.9968608 0.9966364 0.9967673 0.9966898 0.9968049 0.9968184
##  [610] 0.9967486 0.9967463 0.9967394 0.9967647 0.9968172 0.9968159 0.9967480
##  [617] 0.9967134 0.9967674 0.9968071 0.9966878 0.9967433 0.9968311 0.9966660
##  [624] 0.9967827 0.9968543 0.9967522 0.9967793 0.9966758 0.9967532 0.9967118
##  [631] 0.9968329 0.9967167 0.9967296 0.9966920 0.9967664 0.9966677 0.9967094
##  [638] 0.9968450 0.9967034 0.9967453 0.9967777 0.9967364 0.9967558 0.9967228
##  [645] 0.9968198 0.9967331 0.9968028 0.9967166 0.9967938 0.9967391 0.9967626
##  [652] 0.9967306 0.9966934 0.9967729 0.9967377 0.9967162 0.9967149 0.9967293
##  [659] 0.9967658 0.9968494 0.9967623 0.9967748 0.9967124 0.9967916 0.9967091
##  [666] 0.9966911 0.9966569 0.9967486 0.9967620 0.9968087 0.9966916 0.9968017
##  [673] 0.9966739 0.9966958 0.9966735 0.9968149 0.9967934 0.9966950 0.9966689
##  [680] 0.9967345 0.9966383 0.9967809 0.9966920 0.9968042 0.9967975 0.9967687
##  [687] 0.9967160 0.9967514 0.9967021 0.9967186 0.9966911 0.9967350 0.9967650
##  [694] 0.9966706 0.9966951 0.9968002 0.9966536 0.9968113 0.9967808 0.9967786
##  [701] 0.9967344 0.9968120 0.9968284 0.9967796 0.9967399 0.9968064 0.9967755
##  [708] 0.9967805 0.9967251 0.9967057 0.9967673 0.9967516 0.9967114 0.9967669
##  [715] 0.9967579 0.9967797 0.9968586 0.9967901 0.9967673 0.9967974 0.9966884
##  [722] 0.9967362 0.9967223 0.9967467 0.9967184 0.9968403 0.9967417 0.9967193
##  [729] 0.9968384 0.9967343 0.9967532 0.9968347 0.9967616 0.9967120 0.9967188
##  [736] 0.9968045 0.9967701 0.9967821 0.9967235 0.9966925 0.9967745 0.9967707
##  [743] 0.9967699 0.9967892 0.9968163 0.9967250 0.9967744 0.9966632 0.9968126
##  [750] 0.9967199 0.9966272 0.9967568 0.9967531 0.9967767 0.9968182 0.9967468
##  [757] 0.9967953 0.9967334 0.9967151 0.9968693 0.9967763 0.9967079 0.9968033
##  [764] 0.9967822 0.9967122 0.9966511 0.9967785 0.9967456 0.9967647 0.9966896
##  [771] 0.9967645 0.9967585 0.9967530 0.9967832 0.9967581 0.9967540 0.9967649
##  [778] 0.9967488 0.9967318 0.9966671 0.9967959 0.9966357 0.9967393 0.9967698
##  [785] 0.9968246 0.9967657 0.9967858 0.9967544 0.9968106 0.9968126 0.9968016
##  [792] 0.9967282 0.9966753 0.9967757 0.9967854 0.9966400 0.9966905 0.9966289
##  [799] 0.9967766 0.9966988 0.9967029 0.9967087 0.9967430 0.9967592 0.9967527
##  [806] 0.9967364 0.9967777 0.9967230 0.9967589 0.9968431 0.9967330 0.9967800
##  [813] 0.9967756 0.9967313 0.9967399 0.9967702 0.9967315 0.9967674 0.9967920
##  [820] 0.9967491 0.9966569 0.9967737 0.9966991 0.9966939 0.9967141 0.9968047
##  [827] 0.9967440 0.9967789 0.9967317 0.9967332 0.9967582 0.9966928 0.9967673
##  [834] 0.9968376 0.9966868 0.9967704 0.9967635 0.9967133 0.9968157 0.9967393
##  [841] 0.9967944 0.9968195 0.9967658 0.9968027 0.9967418 0.9967851 0.9966931
##  [848] 0.9967870 0.9968115 0.9967395 0.9967900 0.9967345 0.9967574 0.9966704
##  [855] 0.9967085 0.9967469 0.9967113 0.9966976 0.9967370 0.9967747 0.9968180
##  [862] 0.9967282 0.9967241 0.9967069 0.9967685 0.9968337 0.9967878 0.9967464
##  [869] 0.9968012 0.9967517 0.9967398 0.9967423 0.9967848 0.9967022 0.9967203
##  [876] 0.9967786 0.9967402 0.9967545 0.9967076 0.9967711 0.9967665 0.9967602
##  [883] 0.9966798 0.9967273 0.9967078 0.9967420 0.9966975 0.9967293 0.9966159
##  [890] 0.9967192 0.9967268 0.9967967 0.9967267 0.9967142 0.9967397 0.9967228
##  [897] 0.9967796 0.9967765 0.9967892 0.9968219 0.9967972 0.9967317 0.9968087
##  [904] 0.9968229 0.9967184 0.9967884 0.9967830 0.9967942 0.9966883 0.9967400
##  [911] 0.9967985 0.9967425 0.9967367 0.9968065 0.9967564 0.9968179 0.9968104
##  [918] 0.9967891 0.9966559 0.9967463 0.9967452 0.9968093 0.9966773 0.9967734
##  [925] 0.9967588 0.9967617 0.9967932 0.9967825 0.9968608 0.9968214 0.9967286
##  [932] 0.9968852 0.9967364 0.9967729 0.9967972 0.9967227 0.9967647 0.9967521
##  [939] 0.9967611 0.9967353 0.9968260 0.9967524 0.9967989 0.9967028 0.9966832
##  [946] 0.9967036 0.9967788 0.9967111 0.9967898 0.9967810 0.9966771 0.9967424
##  [953] 0.9968430 0.9966895 0.9967293 0.9968441 0.9967241 0.9968043 0.9967453
##  [960] 0.9966429 0.9966861 0.9966598 0.9967260 0.9967183 0.9967276 0.9967491
##  [967] 0.9967533 0.9968021 0.9967590 0.9967583 0.9966786 0.9967197 0.9967157
##  [974] 0.9967719 0.9967184 0.9967658 0.9966844 0.9967756 0.9968041 0.9967896
##  [981] 0.9967970 0.9967169 0.9967233 0.9967687 0.9967112 0.9966810 0.9966536
##  [988] 0.9966918 0.9967650 0.9967593 0.9967497 0.9967299 0.9967767 0.9966515
##  [995] 0.9967987 0.9967668 0.9967050 0.9967603 0.9967791 0.9967879 0.9967335
## [1002] 0.9966986 0.9966793 0.9967852 0.9967504 0.9966269 0.9967327 0.9967693
## [1009] 0.9967606 0.9967221 0.9967174 0.9967553 0.9966695 0.9968093 0.9967628
## [1016] 0.9967326 0.9966056 0.9967617 0.9966361 0.9968198 0.9967414 0.9968750
## [1023] 0.9967485 0.9967101 0.9966754 0.9968013 0.9967799 0.9967324 0.9967887
## [1030] 0.9967143 0.9967232 0.9967072 0.9967262 0.9967333 0.9967690 0.9966978
## [1037] 0.9967924 0.9967843 0.9967103 0.9967562 0.9967478 0.9967466 0.9967232
## [1044] 0.9968118 0.9966775 0.9967012 0.9967264 0.9967843 0.9967129 0.9967926
## [1051] 0.9968218 0.9966621 0.9967588 0.9967369 0.9966916 0.9968389 0.9967581
## [1058] 0.9967816 0.9967370 0.9968728 0.9967566 0.9967223 0.9967199 0.9967676
## [1065] 0.9968030 0.9967463 0.9967588 0.9968299 0.9966556 0.9966405 0.9966275
## [1072] 0.9967517 0.9967611 0.9966852 0.9967461 0.9967318 0.9967005 0.9967531
## [1079] 0.9968122 0.9967730 0.9967399 0.9966884 0.9967196 0.9968160 0.9966718
## [1086] 0.9967261 0.9967542 0.9967552 0.9967987 0.9968297 0.9967129 0.9967958
## [1093] 0.9967671 0.9967226 0.9966947 0.9967570 0.9967091 0.9966921 0.9967102
## [1100] 0.9966935 0.9966729 0.9966411 0.9967984 0.9967079 0.9968393 0.9967665
## [1107] 0.9967675 0.9967042 0.9967148 0.9967672 0.9968069 0.9967154 0.9967433
## [1114] 0.9967226 0.9967840 0.9967871 0.9966573 0.9967302 0.9967752 0.9966337
## [1121] 0.9967443 0.9967399 0.9967599 0.9966789 0.9966230 0.9967534 0.9966636
## [1128] 0.9968250 0.9967641 0.9968073 0.9967559 0.9967920 0.9967634 0.9968055
## [1135] 0.9967018 0.9967575 0.9968311 0.9967095 0.9966972 0.9967571 0.9967098
## [1142] 0.9967130 0.9967946 0.9968481 0.9967320 0.9968051 0.9967572 0.9967265
## [1149] 0.9967690 0.9967686 0.9967146 0.9967232 0.9967121 0.9967215 0.9967318
## [1156] 0.9967378 0.9967373 0.9967865 0.9967353 0.9967470 0.9967042 0.9968043
## [1163] 0.9967013 0.9966897 0.9967568 0.9967745 0.9968169 0.9967832 0.9967007
## [1170] 0.9967301 0.9966968 0.9967506 0.9967460 0.9967542 0.9968005 0.9967681
## [1177] 0.9967191 0.9967276 0.9967146 0.9966890 0.9967814 0.9967530 0.9967257
## [1184] 0.9967776 0.9967669 0.9967235 0.9967430 0.9967494 0.9967169 0.9967532
## [1191] 0.9967800 0.9967735 0.9966970 0.9968007 0.9966767 0.9968579 0.9968114
## [1198] 0.9967616 0.9967740 0.9966510 0.9967146 0.9967041 0.9967531 0.9967283
## [1205] 0.9968098 0.9967201 0.9967770 0.9967700 0.9967050 0.9967602 0.9967896
## [1212] 0.9967190 0.9967995 0.9968148 0.9968253 0.9967599 0.9967950 0.9967687
## [1219] 0.9966744 0.9967094 0.9967269 0.9967266 0.9967503 0.9967981 0.9967833
## [1226] 0.9967607 0.9967825 0.9967523 0.9967298 0.9966983 0.9967027 0.9967580
## [1233] 0.9967831 0.9967187 0.9968037 0.9967882 0.9967972 0.9967114 0.9967511
## [1240] 0.9967907 0.9967484 0.9967367 0.9967125 0.9967143 0.9967820 0.9967477
## [1247] 0.9967645 0.9968614 0.9966956 0.9966925 0.9967823 0.9967250 0.9967204
## [1254] 0.9967617 0.9967826 0.9966925 0.9966836 0.9967483 0.9967607 0.9967549
## [1261] 0.9966969 0.9967255 0.9967431 0.9967912 0.9967275 0.9967376 0.9967392
## [1268] 0.9966551 0.9967788 0.9967820 0.9967846 0.9967146 0.9967669 0.9967606
## [1275] 0.9968245 0.9967320 0.9966637 0.9967408 0.9967051 0.9967516 0.9968320
## [1282] 0.9967928 0.9967009 0.9967066 0.9966966 0.9967325 0.9967702 0.9966801
## [1289] 0.9966755 0.9967449 0.9967075 0.9968271 0.9967905 0.9967075 0.9967348
## [1296] 0.9967040 0.9967155 0.9967635 0.9967559 0.9967006 0.9968316 0.9966987
## [1303] 0.9967929 0.9967421 0.9967861 0.9967852 0.9967969 0.9967394 0.9968114
## [1310] 0.9967233 0.9966576 0.9967490 0.9967938 0.9967414 0.9967471 0.9967250
## [1317] 0.9966953 0.9966769 0.9967142 0.9966677 0.9967606 0.9966169 0.9967666
## [1324] 0.9967748 0.9967823 0.9968179 0.9967157 0.9966954 0.9967718 0.9967588
## [1331] 0.9967842 0.9968010 0.9967600 0.9967291 0.9967277 0.9968036 0.9968662
## [1338] 0.9967316 0.9968037 0.9967433 0.9967894 0.9967603 0.9967428 0.9967734
## [1345] 0.9967670 0.9968244 0.9966247 0.9967768 0.9966945 0.9967602 0.9968299
## [1352] 0.9967813 0.9968470 0.9967804 0.9966421 0.9968126 0.9967309 0.9967818
## [1359] 0.9967492 0.9967504 0.9967637 0.9967829 0.9968201 0.9967066 0.9968844
## [1366] 0.9967036 0.9967554 0.9967359 0.9967147 0.9966979 0.9967667 0.9967757
## [1373] 0.9968219 0.9967274 0.9966618 0.9967958 0.9966669 0.9966925 0.9968238
## [1380] 0.9967213 0.9968054 0.9966631 0.9966287 0.9967482 0.9966771 0.9967102
## [1387] 0.9967298 0.9967027 0.9967779 0.9967796 0.9967099 0.9967392 0.9966994
## [1394] 0.9966401 0.9967136 0.9966734 0.9967286 0.9967541 0.9968207 0.9968023
## [1401] 0.9967097 0.9967412 0.9967322 0.9968040 0.9968003 0.9966938 0.9967963
## [1408] 0.9967419 0.9967081 0.9968043 0.9967219 0.9967782 0.9968003 0.9967200
## [1415] 0.9968293 0.9968136 0.9966997 0.9967748 0.9967339 0.9968176 0.9966678
## [1422] 0.9966942 0.9967898 0.9967038 0.9967452 0.9967574 0.9967076 0.9967587
## [1429] 0.9967336 0.9967236 0.9967661 0.9966265 0.9967981 0.9967703 0.9968658
## [1436] 0.9967611 0.9967813 0.9967180 0.9968451 0.9967605 0.9967818 0.9967840
## [1443] 0.9967872 0.9967216 0.9968038 0.9967273 0.9967135 0.9967787 0.9967381
## [1450] 0.9968382 0.9968416 0.9967594 0.9966933 0.9967882 0.9967463 0.9968496
## [1457] 0.9967297 0.9967724 0.9967105 0.9967131 0.9968023 0.9967163 0.9967844
## [1464] 0.9967494 0.9967838 0.9967787 0.9967006 0.9967389 0.9968135 0.9967964
## [1471] 0.9967221 0.9967800 0.9967549 0.9967950 0.9968120 0.9968046 0.9967553
## [1478] 0.9967531 0.9968470 0.9968163 0.9967265 0.9966425 0.9966926 0.9967490
## [1485] 0.9967411 0.9967416 0.9967600 0.9966944 0.9967871 0.9966303 0.9967348
## [1492] 0.9967321 0.9967023 0.9966995 0.9967918 0.9967821 0.9966336 0.9967896
## [1499] 0.9968146 0.9967884 0.9967584 0.9967460 0.9967625 0.9967868 0.9966915
## [1506] 0.9967840 0.9967677 0.9968045 0.9967035 0.9967490 0.9967640 0.9966417
## [1513] 0.9967734 0.9968187 0.9966724 0.9967109 0.9967534 0.9968005 0.9967540
## [1520] 0.9966933 0.9967992 0.9967801 0.9967886 0.9968306 0.9967973 0.9966990
## [1527] 0.9966508 0.9966941 0.9967904 0.9967258 0.9966975 0.9968142 0.9967685
## [1534] 0.9967484 0.9967117 0.9967645 0.9967620 0.9967623 0.9967886 0.9967700
## [1541] 0.9968108 0.9967938 0.9968439 0.9967944 0.9967434 0.9967207 0.9967304
## [1548] 0.9967369 0.9967683 0.9967754 0.9967164 0.9967887 0.9967832 0.9966369
## [1555] 0.9967379 0.9966758 0.9967751 0.9967503 0.9967964 0.9967512 0.9967474
## [1562] 0.9967026 0.9967661 0.9967153 0.9967208 0.9968069 0.9967156 0.9967224
## [1569] 0.9967022 0.9967681 0.9968132 0.9967012 0.9968122 0.9967626 0.9968385
## [1576] 0.9967770 0.9967437 0.9967562 0.9967438 0.9966932 0.9966856 0.9966415
## [1583] 0.9967515 0.9967979 0.9967340 0.9967183 0.9967143 0.9968709 0.9966690
## [1590] 0.9966870 0.9967497 0.9967459 0.9967927 0.9967731 0.9966976 0.9967919
## [1597] 0.9967561 0.9967574 0.9966648 0.9967231 0.9968306 0.9967555 0.9967940
## [1604] 0.9967022 0.9967646 0.9968272 0.9967828 0.9967467 0.9967475 0.9967593
## [1611] 0.9967607 0.9967824 0.9966494 0.9966543 0.9966152 0.9967633 0.9966827
## [1618] 0.9967256 0.9967081 0.9967208 0.9967212 0.9966797 0.9967994 0.9967883
## [1625] 0.9966574 0.9967431 0.9967001 0.9967067 0.9967371 0.9967415 0.9966675
## [1632] 0.9967144 0.9967486 0.9967318 0.9967312 0.9967928 0.9967587 0.9967661
## [1639] 0.9967526 0.9967428 0.9967158 0.9968337 0.9967501 0.9966930 0.9966961
## [1646] 0.9968403 0.9966998 0.9967474 0.9967658 0.9966986 0.9966960 0.9967444
## [1653] 0.9967401 0.9967546 0.9967821 0.9966935 0.9967163 0.9967907 0.9967532
## [1660] 0.9967159 0.9967551 0.9967890 0.9967612 0.9967755 0.9967527 0.9967344
## [1667] 0.9966781 0.9967890 0.9967287 0.9967782 0.9967763 0.9967006 0.9967587
## [1674] 0.9966984 0.9966848 0.9967903 0.9967385 0.9967777 0.9967892 0.9968089
## [1681] 0.9967454 0.9967212 0.9967459 0.9966688 0.9966568 0.9967076 0.9967991
## [1688] 0.9968207 0.9968773 0.9968207 0.9966942 0.9967238 0.9968129 0.9967055
## [1695] 0.9967597 0.9967719 0.9967369 0.9967629 0.9967523 0.9968469 0.9967555
## [1702] 0.9966992 0.9967513 0.9966692 0.9967574 0.9967321 0.9967522 0.9967832
## [1709] 0.9967414 0.9966896 0.9967500 0.9967708 0.9966910 0.9968588 0.9967487
## [1716] 0.9967308 0.9968194 0.9967836 0.9967242 0.9966092 0.9967252 0.9966891
## [1723] 0.9968238 0.9967105 0.9967998 0.9966827 0.9967493 0.9967999 0.9966847
## [1730] 0.9967443 0.9967678 0.9967618 0.9967316 0.9967703 0.9967447 0.9967393
## [1737] 0.9967400 0.9967834 0.9967132 0.9967410 0.9967687 0.9966873 0.9967262
## [1744] 0.9967170 0.9967045 0.9966723 0.9967560 0.9968250 0.9967979 0.9966702
## [1751] 0.9967474 0.9967418 0.9968155 0.9967545 0.9966970 0.9967283 0.9967519
## [1758] 0.9968249 0.9967137 0.9968430 0.9967998 0.9966523 0.9967248 0.9967353
## [1765] 0.9967755 0.9967440 0.9967861 0.9967761 0.9966778 0.9966827 0.9966905
## [1772] 0.9967079 0.9967207 0.9967732 0.9967606 0.9967623 0.9967700 0.9967632
## [1779] 0.9967194 0.9969141 0.9966947 0.9966928 0.9968166 0.9967179 0.9967899
## [1786] 0.9967510 0.9967678 0.9967208 0.9966785 0.9968061 0.9967995 0.9967227
## [1793] 0.9967921 0.9967502 0.9968312 0.9967492 0.9967338 0.9967130 0.9967166
## [1800] 0.9967794 0.9967452 0.9967209 0.9967212 0.9967439 0.9967249 0.9967942
## [1807] 0.9967133 0.9967125 0.9966936 0.9968204 0.9967595 0.9967740 0.9967990
## [1814] 0.9966684 0.9967689 0.9967409 0.9967466 0.9967935 0.9967335 0.9967256
## [1821] 0.9968524 0.9967517 0.9967717 0.9967519 0.9967490 0.9968093 0.9967263
## [1828] 0.9967292 0.9967819 0.9967583 0.9966405 0.9968008 0.9966737 0.9966961
## [1835] 0.9966692 0.9967211 0.9967626 0.9967573 0.9967539 0.9967179 0.9967582
## [1842] 0.9966521 0.9967185 0.9967750 0.9967965 0.9966428 0.9967347 0.9967731
## [1849] 0.9967384 0.9967470 0.9967628 0.9967170 0.9968174 0.9967710 0.9968059
## [1856] 0.9967422 0.9967478 0.9967674 0.9967257 0.9967045 0.9968115 0.9967140
## [1863] 0.9967831 0.9967286 0.9967044 0.9966318 0.9966943 0.9967248 0.9967086
## [1870] 0.9967588 0.9967403 0.9966876 0.9967089 0.9967209 0.9967202 0.9967769
## [1877] 0.9967432 0.9968372 0.9967748 0.9967785 0.9967369 0.9967281 0.9967435
## [1884] 0.9967130 0.9966608 0.9966900 0.9967332 0.9967816 0.9966734 0.9967333
## [1891] 0.9967740 0.9968012 0.9968319 0.9966996 0.9967509 0.9968273 0.9967346
## [1898] 0.9967497 0.9968061 0.9968340 0.9966995 0.9967006 0.9967468 0.9967982
## [1905] 0.9967530 0.9967935 0.9966909 0.9967711 0.9967613 0.9966058 0.9968121
## [1912] 0.9967727 0.9967818 0.9967043 0.9967350 0.9967381 0.9966703 0.9967030
## [1919] 0.9967676 0.9968009 0.9967420 0.9966459 0.9967418 0.9966258 0.9967588
## [1926] 0.9968007 0.9967598 0.9967400 0.9967214 0.9968057 0.9967575 0.9966882
## [1933] 0.9966707 0.9967339 0.9968242 0.9967294 0.9966783 0.9966955 0.9967246
## [1940] 0.9967329 0.9967248 0.9966796 0.9967147 0.9967417 0.9967312 0.9967210
## [1947] 0.9966375 0.9966530 0.9967254 0.9967090 0.9968180 0.9967696 0.9968606
## [1954] 0.9967614 0.9967695 0.9967909 0.9967947 0.9967553 0.9967864 0.9966743
## [1961] 0.9967231 0.9966951 0.9967482 0.9968261 0.9967920 0.9968029 0.9967634
## [1968] 0.9967651 0.9967238 0.9967391 0.9967535 0.9967744 0.9968058 0.9968187
## [1975] 0.9966725 0.9966937 0.9968278 0.9967461 0.9967988 0.9967071 0.9967205
## [1982] 0.9967345 0.9967529 0.9967473 0.9966869 0.9967675 0.9967679 0.9967353
## [1989] 0.9967454 0.9967846 0.9967279 0.9967726 0.9966985 0.9967032 0.9967704
## [1996] 0.9967610 0.9966766 0.9967355 0.9967662 0.9968241
upper <- mu.density.set + 2*(var.density.set)
upper
##    [1] 0.9967614 0.9967688 0.9967200 0.9967184 0.9966774 0.9968033 0.9967664
##    [8] 0.9969063 0.9967110 0.9967694 0.9967769 0.9967441 0.9966910 0.9966458
##   [15] 0.9967292 0.9967319 0.9968028 0.9967832 0.9967664 0.9967837 0.9967461
##   [22] 0.9967228 0.9968042 0.9968051 0.9967747 0.9967699 0.9967297 0.9967589
##   [29] 0.9967398 0.9967728 0.9967669 0.9967297 0.9968006 0.9966556 0.9967871
##   [36] 0.9967340 0.9967553 0.9967121 0.9966997 0.9967970 0.9967840 0.9966496
##   [43] 0.9967391 0.9966814 0.9967970 0.9967209 0.9967375 0.9967936 0.9966770
##   [50] 0.9966683 0.9967974 0.9967772 0.9967315 0.9967526 0.9967868 0.9967735
##   [57] 0.9967705 0.9967185 0.9967267 0.9967440 0.9967231 0.9967660 0.9966411
##   [64] 0.9967141 0.9968102 0.9968279 0.9967155 0.9966315 0.9967939 0.9967784
##   [71] 0.9968183 0.9967855 0.9967917 0.9967928 0.9967744 0.9968044 0.9967147
##   [78] 0.9967671 0.9967319 0.9966838 0.9967308 0.9966792 0.9967779 0.9967641
##   [85] 0.9966817 0.9967438 0.9968048 0.9967552 0.9967581 0.9967986 0.9967210
##   [92] 0.9967533 0.9967243 0.9967522 0.9967296 0.9967594 0.9968117 0.9967937
##   [99] 0.9967664 0.9967764 0.9966496 0.9966964 0.9968137 0.9966461 0.9966585
##  [106] 0.9967137 0.9966859 0.9967904 0.9968072 0.9967050 0.9967695 0.9967940
##  [113] 0.9967145 0.9966570 0.9966732 0.9967785 0.9966872 0.9967497 0.9967482
##  [120] 0.9967467 0.9966894 0.9966966 0.9967329 0.9967671 0.9967034 0.9967243
##  [127] 0.9968035 0.9967500 0.9967266 0.9966547 0.9967175 0.9967157 0.9967034
##  [134] 0.9967792 0.9967876 0.9967080 0.9967640 0.9967867 0.9966659 0.9967986
##  [141] 0.9967306 0.9967365 0.9967545 0.9967497 0.9968205 0.9966970 0.9967762
##  [148] 0.9967593 0.9966969 0.9966759 0.9967403 0.9967115 0.9967716 0.9967314
##  [155] 0.9967551 0.9966559 0.9968233 0.9967621 0.9967279 0.9966588 0.9966766
##  [162] 0.9967685 0.9966974 0.9967802 0.9967727 0.9967752 0.9966979 0.9967078
##  [169] 0.9967362 0.9967735 0.9967576 0.9967394 0.9967322 0.9966803 0.9966296
##  [176] 0.9967764 0.9967857 0.9967560 0.9967031 0.9967819 0.9968294 0.9966561
##  [183] 0.9967771 0.9967328 0.9968088 0.9967011 0.9967178 0.9967470 0.9966738
##  [190] 0.9967674 0.9967272 0.9967630 0.9967474 0.9967917 0.9967416 0.9968388
##  [197] 0.9966857 0.9967679 0.9967394 0.9967022 0.9967689 0.9967933 0.9966535
##  [204] 0.9968052 0.9968337 0.9967627 0.9968279 0.9967631 0.9967390 0.9966332
##  [211] 0.9966651 0.9966513 0.9967207 0.9967713 0.9967655 0.9967832 0.9967940
##  [218] 0.9967318 0.9967377 0.9967062 0.9967155 0.9967696 0.9966742 0.9967980
##  [225] 0.9967477 0.9967851 0.9968713 0.9967464 0.9967163 0.9967415 0.9966662
##  [232] 0.9967987 0.9967436 0.9967775 0.9966972 0.9967862 0.9967649 0.9967987
##  [239] 0.9967509 0.9967483 0.9967587 0.9967645 0.9967748 0.9967275 0.9967251
##  [246] 0.9967291 0.9967095 0.9967703 0.9967573 0.9967500 0.9967472 0.9967465
##  [253] 0.9968509 0.9968158 0.9967352 0.9967784 0.9967562 0.9967460 0.9968394
##  [260] 0.9967138 0.9967533 0.9967990 0.9967152 0.9967905 0.9967174 0.9966944
##  [267] 0.9967650 0.9967566 0.9967456 0.9967158 0.9966441 0.9967265 0.9967921
##  [274] 0.9966791 0.9967655 0.9967104 0.9967020 0.9967846 0.9967204 0.9968129
##  [281] 0.9967488 0.9967594 0.9967108 0.9967639 0.9967387 0.9967932 0.9967271
##  [288] 0.9968367 0.9966752 0.9967420 0.9966847 0.9967851 0.9966877 0.9968178
##  [295] 0.9968031 0.9966464 0.9967853 0.9967157 0.9967545 0.9967902 0.9966879
##  [302] 0.9967632 0.9966850 0.9967913 0.9967881 0.9967567 0.9967303 0.9967814
##  [309] 0.9967302 0.9966989 0.9967194 0.9967951 0.9967338 0.9968033 0.9968194
##  [316] 0.9967114 0.9967791 0.9966916 0.9967353 0.9968214 0.9967433 0.9967489
##  [323] 0.9968361 0.9967651 0.9966776 0.9966330 0.9967216 0.9967768 0.9967225
##  [330] 0.9967127 0.9967218 0.9967820 0.9967473 0.9967302 0.9967516 0.9967801
##  [337] 0.9967850 0.9967650 0.9967633 0.9967254 0.9967237 0.9967788 0.9967162
##  [344] 0.9968131 0.9967368 0.9967929 0.9967867 0.9967126 0.9968523 0.9966773
##  [351] 0.9967075 0.9967252 0.9967205 0.9968054 0.9967705 0.9967634 0.9967317
##  [358] 0.9967487 0.9967899 0.9968047 0.9967498 0.9967803 0.9966997 0.9968257
##  [365] 0.9968066 0.9967868 0.9967002 0.9966894 0.9968003 0.9966602 0.9967082
##  [372] 0.9968193 0.9967190 0.9967611 0.9967367 0.9966981 0.9967060 0.9966292
##  [379] 0.9966920 0.9967886 0.9967357 0.9967491 0.9967231 0.9967738 0.9967794
##  [386] 0.9967725 0.9967731 0.9967050 0.9967214 0.9967826 0.9967166 0.9966915
##  [393] 0.9967048 0.9967541 0.9966966 0.9967567 0.9967401 0.9967721 0.9967127
##  [400] 0.9967431 0.9968030 0.9967539 0.9968191 0.9968106 0.9967462 0.9967429
##  [407] 0.9967199 0.9967024 0.9968097 0.9966804 0.9968758 0.9968366 0.9966906
##  [414] 0.9967503 0.9966912 0.9968675 0.9967194 0.9967729 0.9966922 0.9966241
##  [421] 0.9968178 0.9966528 0.9967557 0.9967218 0.9966910 0.9967433 0.9967688
##  [428] 0.9968247 0.9967140 0.9968269 0.9967063 0.9968519 0.9967086 0.9967210
##  [435] 0.9967568 0.9967444 0.9966496 0.9967161 0.9967297 0.9966547 0.9967421
##  [442] 0.9967274 0.9966922 0.9968385 0.9967344 0.9967355 0.9967361 0.9968839
##  [449] 0.9967760 0.9967057 0.9967618 0.9967095 0.9967367 0.9967147 0.9968040
##  [456] 0.9967551 0.9967393 0.9967179 0.9967131 0.9967376 0.9967590 0.9967423
##  [463] 0.9967112 0.9967122 0.9967321 0.9967810 0.9967767 0.9968999 0.9967711
##  [470] 0.9967397 0.9967644 0.9968147 0.9967249 0.9966674 0.9967102 0.9966737
##  [477] 0.9968181 0.9967059 0.9967883 0.9968140 0.9967180 0.9967907 0.9968008
##  [484] 0.9967759 0.9968112 0.9967554 0.9967670 0.9967328 0.9968031 0.9967081
##  [491] 0.9966770 0.9967791 0.9967253 0.9967532 0.9967796 0.9967341 0.9967078
##  [498] 0.9967325 0.9967624 0.9966741 0.9967272 0.9967594 0.9967719 0.9967724
##  [505] 0.9967327 0.9968090 0.9967543 0.9967091 0.9967086 0.9967574 0.9966818
##  [512] 0.9968104 0.9967751 0.9967437 0.9967119 0.9966114 0.9967485 0.9967346
##  [519] 0.9967544 0.9967356 0.9967942 0.9967666 0.9967557 0.9967458 0.9967195
##  [526] 0.9967942 0.9966714 0.9966841 0.9967162 0.9967367 0.9966805 0.9967336
##  [533] 0.9968055 0.9967622 0.9967301 0.9967418 0.9966896 0.9966351 0.9967578
##  [540] 0.9967603 0.9967121 0.9968117 0.9967280 0.9967776 0.9967313 0.9967283
##  [547] 0.9967080 0.9968100 0.9967606 0.9967211 0.9967487 0.9967265 0.9966914
##  [554] 0.9968036 0.9967193 0.9967046 0.9967370 0.9968220 0.9968761 0.9967370
##  [561] 0.9967513 0.9966937 0.9967315 0.9967447 0.9967409 0.9966787 0.9967257
##  [568] 0.9967252 0.9967367 0.9967718 0.9968455 0.9967155 0.9967487 0.9967493
##  [575] 0.9967004 0.9967129 0.9967589 0.9968185 0.9966847 0.9967466 0.9967491
##  [582] 0.9967393 0.9967203 0.9967195 0.9968253 0.9966861 0.9968455 0.9967509
##  [589] 0.9967470 0.9966972 0.9967634 0.9967199 0.9967259 0.9966880 0.9967602
##  [596] 0.9967662 0.9966214 0.9967244 0.9967448 0.9967650 0.9967818 0.9967777
##  [603] 0.9967263 0.9968608 0.9966365 0.9967673 0.9966898 0.9968049 0.9968184
##  [610] 0.9967486 0.9967463 0.9967394 0.9967647 0.9968173 0.9968159 0.9967480
##  [617] 0.9967135 0.9967674 0.9968071 0.9966878 0.9967433 0.9968311 0.9966660
##  [624] 0.9967827 0.9968543 0.9967522 0.9967793 0.9966758 0.9967533 0.9967118
##  [631] 0.9968329 0.9967167 0.9967296 0.9966920 0.9967664 0.9966677 0.9967094
##  [638] 0.9968450 0.9967034 0.9967453 0.9967777 0.9967365 0.9967558 0.9967228
##  [645] 0.9968198 0.9967331 0.9968028 0.9967166 0.9967938 0.9967391 0.9967627
##  [652] 0.9967306 0.9966934 0.9967729 0.9967377 0.9967162 0.9967149 0.9967293
##  [659] 0.9967658 0.9968494 0.9967624 0.9967748 0.9967124 0.9967917 0.9967091
##  [666] 0.9966911 0.9966569 0.9967486 0.9967620 0.9968087 0.9966916 0.9968017
##  [673] 0.9966739 0.9966959 0.9966735 0.9968149 0.9967934 0.9966950 0.9966689
##  [680] 0.9967345 0.9966384 0.9967810 0.9966920 0.9968042 0.9967975 0.9967687
##  [687] 0.9967160 0.9967514 0.9967021 0.9967186 0.9966912 0.9967351 0.9967650
##  [694] 0.9966706 0.9966951 0.9968003 0.9966536 0.9968113 0.9967808 0.9967786
##  [701] 0.9967344 0.9968120 0.9968284 0.9967796 0.9967399 0.9968064 0.9967755
##  [708] 0.9967805 0.9967251 0.9967057 0.9967673 0.9967516 0.9967114 0.9967669
##  [715] 0.9967579 0.9967797 0.9968586 0.9967901 0.9967673 0.9967974 0.9966884
##  [722] 0.9967362 0.9967224 0.9967467 0.9967184 0.9968403 0.9967417 0.9967194
##  [729] 0.9968384 0.9967343 0.9967532 0.9968347 0.9967617 0.9967120 0.9967188
##  [736] 0.9968045 0.9967701 0.9967821 0.9967235 0.9966925 0.9967745 0.9967707
##  [743] 0.9967699 0.9967892 0.9968163 0.9967250 0.9967744 0.9966632 0.9968126
##  [750] 0.9967199 0.9966272 0.9967568 0.9967531 0.9967767 0.9968182 0.9967469
##  [757] 0.9967953 0.9967335 0.9967151 0.9968693 0.9967763 0.9967079 0.9968033
##  [764] 0.9967822 0.9967122 0.9966511 0.9967785 0.9967457 0.9967647 0.9966896
##  [771] 0.9967645 0.9967585 0.9967530 0.9967832 0.9967581 0.9967540 0.9967649
##  [778] 0.9967488 0.9967318 0.9966671 0.9967959 0.9966357 0.9967393 0.9967698
##  [785] 0.9968246 0.9967657 0.9967858 0.9967544 0.9968106 0.9968126 0.9968016
##  [792] 0.9967282 0.9966753 0.9967757 0.9967854 0.9966400 0.9966905 0.9966289
##  [799] 0.9967766 0.9966988 0.9967029 0.9967087 0.9967430 0.9967592 0.9967527
##  [806] 0.9967364 0.9967777 0.9967230 0.9967589 0.9968431 0.9967330 0.9967800
##  [813] 0.9967757 0.9967313 0.9967400 0.9967702 0.9967315 0.9967675 0.9967920
##  [820] 0.9967491 0.9966569 0.9967737 0.9966991 0.9966939 0.9967141 0.9968047
##  [827] 0.9967440 0.9967789 0.9967317 0.9967332 0.9967582 0.9966929 0.9967673
##  [834] 0.9968376 0.9966868 0.9967704 0.9967635 0.9967133 0.9968158 0.9967393
##  [841] 0.9967944 0.9968195 0.9967659 0.9968027 0.9967418 0.9967851 0.9966931
##  [848] 0.9967870 0.9968115 0.9967395 0.9967900 0.9967345 0.9967574 0.9966704
##  [855] 0.9967085 0.9967469 0.9967113 0.9966976 0.9967370 0.9967747 0.9968180
##  [862] 0.9967282 0.9967241 0.9967069 0.9967685 0.9968337 0.9967878 0.9967464
##  [869] 0.9968012 0.9967517 0.9967398 0.9967423 0.9967848 0.9967022 0.9967203
##  [876] 0.9967786 0.9967402 0.9967545 0.9967076 0.9967711 0.9967665 0.9967602
##  [883] 0.9966798 0.9967273 0.9967078 0.9967420 0.9966975 0.9967293 0.9966159
##  [890] 0.9967192 0.9967268 0.9967967 0.9967268 0.9967142 0.9967397 0.9967228
##  [897] 0.9967796 0.9967765 0.9967892 0.9968219 0.9967972 0.9967317 0.9968087
##  [904] 0.9968229 0.9967184 0.9967884 0.9967830 0.9967942 0.9966883 0.9967400
##  [911] 0.9967986 0.9967425 0.9967367 0.9968065 0.9967564 0.9968179 0.9968104
##  [918] 0.9967892 0.9966559 0.9967463 0.9967452 0.9968093 0.9966774 0.9967734
##  [925] 0.9967588 0.9967617 0.9967932 0.9967825 0.9968608 0.9968214 0.9967286
##  [932] 0.9968852 0.9967364 0.9967729 0.9967972 0.9967227 0.9967647 0.9967521
##  [939] 0.9967612 0.9967353 0.9968260 0.9967524 0.9967989 0.9967028 0.9966832
##  [946] 0.9967036 0.9967788 0.9967111 0.9967898 0.9967810 0.9966771 0.9967424
##  [953] 0.9968430 0.9966895 0.9967293 0.9968441 0.9967241 0.9968043 0.9967453
##  [960] 0.9966429 0.9966861 0.9966598 0.9967260 0.9967183 0.9967276 0.9967491
##  [967] 0.9967533 0.9968021 0.9967590 0.9967583 0.9966786 0.9967197 0.9967157
##  [974] 0.9967719 0.9967184 0.9967658 0.9966844 0.9967756 0.9968042 0.9967896
##  [981] 0.9967970 0.9967169 0.9967233 0.9967687 0.9967112 0.9966810 0.9966537
##  [988] 0.9966918 0.9967651 0.9967593 0.9967497 0.9967299 0.9967767 0.9966515
##  [995] 0.9967987 0.9967668 0.9967050 0.9967603 0.9967791 0.9967879 0.9967335
## [1002] 0.9966986 0.9966793 0.9967852 0.9967504 0.9966269 0.9967327 0.9967693
## [1009] 0.9967606 0.9967221 0.9967174 0.9967553 0.9966695 0.9968093 0.9967628
## [1016] 0.9967326 0.9966056 0.9967617 0.9966361 0.9968198 0.9967414 0.9968750
## [1023] 0.9967485 0.9967101 0.9966754 0.9968013 0.9967799 0.9967324 0.9967887
## [1030] 0.9967143 0.9967232 0.9967072 0.9967262 0.9967333 0.9967690 0.9966978
## [1037] 0.9967924 0.9967843 0.9967103 0.9967563 0.9967478 0.9967466 0.9967232
## [1044] 0.9968118 0.9966775 0.9967012 0.9967264 0.9967843 0.9967129 0.9967926
## [1051] 0.9968218 0.9966621 0.9967588 0.9967369 0.9966916 0.9968389 0.9967581
## [1058] 0.9967816 0.9967370 0.9968729 0.9967566 0.9967223 0.9967200 0.9967676
## [1065] 0.9968030 0.9967463 0.9967588 0.9968299 0.9966556 0.9966405 0.9966275
## [1072] 0.9967517 0.9967611 0.9966853 0.9967461 0.9967318 0.9967005 0.9967532
## [1079] 0.9968122 0.9967730 0.9967399 0.9966884 0.9967196 0.9968160 0.9966718
## [1086] 0.9967261 0.9967543 0.9967553 0.9967987 0.9968297 0.9967129 0.9967958
## [1093] 0.9967671 0.9967226 0.9966947 0.9967571 0.9967091 0.9966921 0.9967102
## [1100] 0.9966935 0.9966729 0.9966411 0.9967984 0.9967079 0.9968393 0.9967665
## [1107] 0.9967675 0.9967042 0.9967148 0.9967672 0.9968069 0.9967154 0.9967433
## [1114] 0.9967226 0.9967840 0.9967871 0.9966573 0.9967302 0.9967752 0.9966337
## [1121] 0.9967443 0.9967399 0.9967600 0.9966789 0.9966230 0.9967535 0.9966636
## [1128] 0.9968251 0.9967641 0.9968073 0.9967559 0.9967920 0.9967634 0.9968055
## [1135] 0.9967018 0.9967575 0.9968311 0.9967095 0.9966972 0.9967571 0.9967098
## [1142] 0.9967130 0.9967946 0.9968482 0.9967320 0.9968051 0.9967572 0.9967265
## [1149] 0.9967690 0.9967686 0.9967146 0.9967232 0.9967122 0.9967215 0.9967318
## [1156] 0.9967378 0.9967373 0.9967865 0.9967354 0.9967470 0.9967042 0.9968043
## [1163] 0.9967013 0.9966897 0.9967568 0.9967745 0.9968169 0.9967833 0.9967007
## [1170] 0.9967301 0.9966968 0.9967506 0.9967461 0.9967542 0.9968005 0.9967681
## [1177] 0.9967191 0.9967276 0.9967146 0.9966890 0.9967814 0.9967530 0.9967257
## [1184] 0.9967776 0.9967669 0.9967235 0.9967430 0.9967494 0.9967170 0.9967532
## [1191] 0.9967800 0.9967735 0.9966970 0.9968007 0.9966768 0.9968579 0.9968114
## [1198] 0.9967616 0.9967740 0.9966510 0.9967146 0.9967041 0.9967531 0.9967283
## [1205] 0.9968098 0.9967201 0.9967770 0.9967700 0.9967050 0.9967602 0.9967896
## [1212] 0.9967190 0.9967995 0.9968149 0.9968253 0.9967599 0.9967951 0.9967687
## [1219] 0.9966745 0.9967094 0.9967269 0.9967266 0.9967503 0.9967981 0.9967833
## [1226] 0.9967607 0.9967825 0.9967523 0.9967298 0.9966983 0.9967028 0.9967580
## [1233] 0.9967831 0.9967187 0.9968038 0.9967882 0.9967972 0.9967114 0.9967511
## [1240] 0.9967907 0.9967484 0.9967367 0.9967125 0.9967143 0.9967820 0.9967477
## [1247] 0.9967646 0.9968614 0.9966956 0.9966925 0.9967823 0.9967251 0.9967204
## [1254] 0.9967617 0.9967826 0.9966925 0.9966836 0.9967483 0.9967607 0.9967549
## [1261] 0.9966969 0.9967255 0.9967431 0.9967913 0.9967275 0.9967376 0.9967392
## [1268] 0.9966552 0.9967788 0.9967821 0.9967846 0.9967146 0.9967669 0.9967606
## [1275] 0.9968245 0.9967320 0.9966637 0.9967408 0.9967051 0.9967516 0.9968320
## [1282] 0.9967928 0.9967009 0.9967066 0.9966966 0.9967325 0.9967702 0.9966801
## [1289] 0.9966755 0.9967449 0.9967075 0.9968271 0.9967905 0.9967075 0.9967348
## [1296] 0.9967040 0.9967156 0.9967635 0.9967560 0.9967006 0.9968316 0.9966987
## [1303] 0.9967929 0.9967421 0.9967861 0.9967852 0.9967969 0.9967394 0.9968114
## [1310] 0.9967233 0.9966576 0.9967490 0.9967938 0.9967414 0.9967471 0.9967251
## [1317] 0.9966953 0.9966769 0.9967142 0.9966677 0.9967607 0.9966169 0.9967666
## [1324] 0.9967748 0.9967823 0.9968179 0.9967157 0.9966954 0.9967718 0.9967588
## [1331] 0.9967842 0.9968010 0.9967600 0.9967291 0.9967277 0.9968036 0.9968662
## [1338] 0.9967316 0.9968037 0.9967433 0.9967894 0.9967603 0.9967428 0.9967734
## [1345] 0.9967671 0.9968245 0.9966248 0.9967769 0.9966945 0.9967602 0.9968299
## [1352] 0.9967813 0.9968471 0.9967804 0.9966421 0.9968126 0.9967309 0.9967818
## [1359] 0.9967492 0.9967504 0.9967637 0.9967829 0.9968201 0.9967066 0.9968844
## [1366] 0.9967036 0.9967555 0.9967359 0.9967147 0.9966979 0.9967667 0.9967757
## [1373] 0.9968219 0.9967274 0.9966618 0.9967958 0.9966669 0.9966925 0.9968238
## [1380] 0.9967213 0.9968054 0.9966631 0.9966287 0.9967482 0.9966771 0.9967102
## [1387] 0.9967298 0.9967027 0.9967779 0.9967796 0.9967099 0.9967392 0.9966994
## [1394] 0.9966401 0.9967136 0.9966734 0.9967286 0.9967541 0.9968207 0.9968023
## [1401] 0.9967097 0.9967412 0.9967322 0.9968040 0.9968003 0.9966938 0.9967964
## [1408] 0.9967420 0.9967081 0.9968043 0.9967219 0.9967782 0.9968003 0.9967201
## [1415] 0.9968293 0.9968136 0.9966997 0.9967748 0.9967340 0.9968176 0.9966678
## [1422] 0.9966942 0.9967898 0.9967038 0.9967452 0.9967574 0.9967076 0.9967587
## [1429] 0.9967336 0.9967236 0.9967661 0.9966265 0.9967981 0.9967703 0.9968658
## [1436] 0.9967611 0.9967813 0.9967180 0.9968452 0.9967605 0.9967818 0.9967840
## [1443] 0.9967873 0.9967216 0.9968038 0.9967273 0.9967135 0.9967787 0.9967381
## [1450] 0.9968382 0.9968416 0.9967594 0.9966933 0.9967882 0.9967463 0.9968496
## [1457] 0.9967297 0.9967724 0.9967105 0.9967131 0.9968023 0.9967163 0.9967844
## [1464] 0.9967494 0.9967838 0.9967787 0.9967006 0.9967389 0.9968135 0.9967964
## [1471] 0.9967221 0.9967800 0.9967549 0.9967950 0.9968120 0.9968046 0.9967553
## [1478] 0.9967532 0.9968470 0.9968163 0.9967266 0.9966425 0.9966926 0.9967490
## [1485] 0.9967411 0.9967416 0.9967600 0.9966944 0.9967871 0.9966303 0.9967348
## [1492] 0.9967321 0.9967023 0.9966995 0.9967918 0.9967821 0.9966336 0.9967896
## [1499] 0.9968146 0.9967884 0.9967585 0.9967460 0.9967625 0.9967868 0.9966915
## [1506] 0.9967841 0.9967677 0.9968045 0.9967035 0.9967490 0.9967640 0.9966417
## [1513] 0.9967734 0.9968187 0.9966724 0.9967109 0.9967534 0.9968005 0.9967540
## [1520] 0.9966933 0.9967992 0.9967801 0.9967886 0.9968306 0.9967973 0.9966990
## [1527] 0.9966508 0.9966941 0.9967904 0.9967258 0.9966975 0.9968142 0.9967685
## [1534] 0.9967484 0.9967118 0.9967645 0.9967620 0.9967623 0.9967886 0.9967700
## [1541] 0.9968108 0.9967938 0.9968439 0.9967944 0.9967434 0.9967207 0.9967304
## [1548] 0.9967369 0.9967683 0.9967754 0.9967165 0.9967887 0.9967832 0.9966369
## [1555] 0.9967380 0.9966758 0.9967751 0.9967503 0.9967964 0.9967513 0.9967474
## [1562] 0.9967026 0.9967661 0.9967153 0.9967208 0.9968069 0.9967156 0.9967225
## [1569] 0.9967022 0.9967681 0.9968132 0.9967012 0.9968122 0.9967626 0.9968385
## [1576] 0.9967770 0.9967437 0.9967562 0.9967438 0.9966932 0.9966856 0.9966415
## [1583] 0.9967515 0.9967979 0.9967340 0.9967183 0.9967143 0.9968709 0.9966690
## [1590] 0.9966870 0.9967497 0.9967459 0.9967928 0.9967731 0.9966976 0.9967919
## [1597] 0.9967561 0.9967574 0.9966648 0.9967232 0.9968306 0.9967555 0.9967940
## [1604] 0.9967022 0.9967646 0.9968272 0.9967828 0.9967467 0.9967475 0.9967593
## [1611] 0.9967607 0.9967824 0.9966494 0.9966543 0.9966152 0.9967633 0.9966827
## [1618] 0.9967257 0.9967081 0.9967208 0.9967212 0.9966797 0.9967994 0.9967883
## [1625] 0.9966574 0.9967431 0.9967001 0.9967067 0.9967371 0.9967415 0.9966675
## [1632] 0.9967145 0.9967486 0.9967318 0.9967312 0.9967928 0.9967587 0.9967661
## [1639] 0.9967527 0.9967428 0.9967158 0.9968337 0.9967501 0.9966930 0.9966961
## [1646] 0.9968403 0.9966998 0.9967474 0.9967658 0.9966986 0.9966960 0.9967444
## [1653] 0.9967402 0.9967546 0.9967821 0.9966935 0.9967163 0.9967907 0.9967532
## [1660] 0.9967159 0.9967551 0.9967890 0.9967612 0.9967755 0.9967527 0.9967344
## [1667] 0.9966782 0.9967890 0.9967287 0.9967782 0.9967764 0.9967006 0.9967587
## [1674] 0.9966984 0.9966848 0.9967903 0.9967385 0.9967777 0.9967892 0.9968089
## [1681] 0.9967454 0.9967212 0.9967459 0.9966688 0.9966568 0.9967076 0.9967991
## [1688] 0.9968207 0.9968773 0.9968207 0.9966942 0.9967238 0.9968129 0.9967055
## [1695] 0.9967597 0.9967719 0.9967369 0.9967629 0.9967523 0.9968469 0.9967555
## [1702] 0.9966992 0.9967513 0.9966692 0.9967574 0.9967321 0.9967522 0.9967832
## [1709] 0.9967414 0.9966896 0.9967501 0.9967708 0.9966910 0.9968588 0.9967487
## [1716] 0.9967308 0.9968194 0.9967836 0.9967242 0.9966092 0.9967252 0.9966891
## [1723] 0.9968239 0.9967105 0.9967998 0.9966827 0.9967493 0.9967999 0.9966848
## [1730] 0.9967443 0.9967678 0.9967618 0.9967316 0.9967703 0.9967447 0.9967393
## [1737] 0.9967400 0.9967835 0.9967133 0.9967410 0.9967687 0.9966873 0.9967262
## [1744] 0.9967171 0.9967045 0.9966723 0.9967560 0.9968250 0.9967979 0.9966702
## [1751] 0.9967474 0.9967418 0.9968155 0.9967545 0.9966970 0.9967283 0.9967519
## [1758] 0.9968249 0.9967137 0.9968430 0.9967998 0.9966523 0.9967248 0.9967353
## [1765] 0.9967756 0.9967440 0.9967861 0.9967762 0.9966778 0.9966827 0.9966905
## [1772] 0.9967079 0.9967207 0.9967732 0.9967606 0.9967623 0.9967700 0.9967632
## [1779] 0.9967194 0.9969141 0.9966947 0.9966928 0.9968166 0.9967180 0.9967899
## [1786] 0.9967510 0.9967678 0.9967208 0.9966785 0.9968061 0.9967995 0.9967227
## [1793] 0.9967921 0.9967502 0.9968312 0.9967492 0.9967338 0.9967130 0.9967166
## [1800] 0.9967794 0.9967452 0.9967209 0.9967212 0.9967439 0.9967249 0.9967943
## [1807] 0.9967134 0.9967125 0.9966936 0.9968204 0.9967595 0.9967740 0.9967990
## [1814] 0.9966684 0.9967689 0.9967409 0.9967466 0.9967935 0.9967335 0.9967256
## [1821] 0.9968524 0.9967517 0.9967717 0.9967519 0.9967490 0.9968093 0.9967263
## [1828] 0.9967292 0.9967819 0.9967583 0.9966405 0.9968008 0.9966737 0.9966961
## [1835] 0.9966692 0.9967211 0.9967626 0.9967573 0.9967540 0.9967179 0.9967582
## [1842] 0.9966521 0.9967185 0.9967750 0.9967965 0.9966428 0.9967347 0.9967731
## [1849] 0.9967384 0.9967471 0.9967628 0.9967170 0.9968174 0.9967710 0.9968059
## [1856] 0.9967422 0.9967478 0.9967674 0.9967257 0.9967045 0.9968115 0.9967140
## [1863] 0.9967831 0.9967286 0.9967045 0.9966318 0.9966943 0.9967248 0.9967086
## [1870] 0.9967588 0.9967403 0.9966876 0.9967089 0.9967209 0.9967202 0.9967769
## [1877] 0.9967432 0.9968372 0.9967749 0.9967785 0.9967369 0.9967281 0.9967435
## [1884] 0.9967130 0.9966608 0.9966900 0.9967333 0.9967816 0.9966734 0.9967333
## [1891] 0.9967740 0.9968012 0.9968319 0.9966996 0.9967509 0.9968273 0.9967347
## [1898] 0.9967497 0.9968061 0.9968340 0.9966995 0.9967006 0.9967468 0.9967982
## [1905] 0.9967530 0.9967935 0.9966909 0.9967711 0.9967613 0.9966058 0.9968121
## [1912] 0.9967727 0.9967818 0.9967043 0.9967351 0.9967381 0.9966703 0.9967031
## [1919] 0.9967676 0.9968009 0.9967420 0.9966459 0.9967419 0.9966258 0.9967588
## [1926] 0.9968007 0.9967598 0.9967400 0.9967214 0.9968057 0.9967575 0.9966882
## [1933] 0.9966707 0.9967339 0.9968242 0.9967294 0.9966783 0.9966955 0.9967247
## [1940] 0.9967329 0.9967248 0.9966796 0.9967147 0.9967417 0.9967312 0.9967210
## [1947] 0.9966375 0.9966530 0.9967254 0.9967090 0.9968180 0.9967696 0.9968606
## [1954] 0.9967614 0.9967695 0.9967909 0.9967947 0.9967553 0.9967864 0.9966743
## [1961] 0.9967231 0.9966951 0.9967482 0.9968262 0.9967920 0.9968029 0.9967634
## [1968] 0.9967651 0.9967238 0.9967391 0.9967535 0.9967744 0.9968058 0.9968187
## [1975] 0.9966725 0.9966937 0.9968278 0.9967461 0.9967988 0.9967071 0.9967205
## [1982] 0.9967345 0.9967529 0.9967473 0.9966869 0.9967675 0.9967679 0.9967353
## [1989] 0.9967454 0.9967846 0.9967279 0.9967726 0.9966985 0.9967032 0.9967704
## [1996] 0.9967610 0.9966766 0.9967355 0.9967662 0.9968241
#red_wine$density - how to make it density
#range of k/n

2. Suppose the population mean of the variable “residual sugar” is μ , a

a. Provide an estimate of μ based on the sample;

mu.sugar <- mean(red_wine$`residual sugar`)
mu.sugar
## [1] 2.538806
The estimate of μ based on the sample mean of “residual sugar” is 2.5388

b. Noting that the sample distribution of “residual sugar” is highly skewed, can we use the CLT to quantify the variability of your estimate? Can we use the CLT to give a 95% confidence interval for μ? If yes, please give your solution. If no, explain why.

hist(red_wine$`residual sugar`)

mu.sugar <- mean(red_wine$`residual sugar`)
mu.sugar
## [1] 2.538806
sd.sugar <- sd(red_wine$`residual sugar`)
sd.sugar
## [1] 1.409928
var.sugar <- sd(red_wine$`residual sugar`)/ sqrt(length(red_wine$`residual sugar`))
var.sugar
## [1] 0.03525922
CLT can be used as the data is normally distributed even if the values are skewed towards right. The variability of the sample of residual sugar is 0.0353
#2b CI

lc.sugar <- mu.sugar - 2*(var.sugar)
lc.sugar
## [1] 2.468287
uc.sugar <- mu.sugar + 2*(var.sugar)
uc.sugar
## [1] 2.609324
ci.sugar.clt <- c(lc.sugar, uc.sugar)
ci.sugar.clt
## [1] 2.468287 2.609324
The upper and lower levels of 95% confidence interval are (2.4683, 2.6093)

c. Use the bootstrap method to do part b. Is the bootstrap confidence interval symmetric? (hint: check the bootstrap distribution; see p. 25-26 in Lecture 4).

mu.sugar.set <- NULL
for (k in 1:2500) {
  sugar.bootstrap <- sample(red_wine$`residual sugar`, size=1599, replace=T)
mu.sugar <- mean(sugar.bootstrap)
mu.sugar.set[k] <- mu.sugar
}

sd(mu.sugar.set)
## [1] 0.03538414
#CI using Bootstrap

conf.q.sugar <- quantile(mu.density.set, probs=c(0.025, 0.975))
conf.q.sugar
##      2.5%     97.5% 
## 0.9966464 0.9968388
The variability using Bootstrap method is almost equal to the variability calculated using CLT

3. We classify those wines as “excellent” if their rating is at least 7. Suppose the population proportion of excellent wines is p. Do the following:

a. Use the CLT to derive a 95% confidence interval for p;

red_wine$excellent <- as.numeric(red_wine$quality > 6)

p <- mean(red_wine$excellent)
p
## [1] 0.1357098
var.excellent <- sqrt(p*(1 - p) / length(red_wine$excellent))
var.excellent
## [1] 0.008564681
lc.p <- p - 2*(var.excellent)
lc.p
## [1] 0.1185805
uc.p <- p + 2*(var.excellent)
uc.p
## [1] 0.1528392

b. Use the bootstrap method to derive a 95% confidence interval for p;

bootstrap_p.set <- NULL
for (k in 1:2500) {
  p.bootstrap <- sample(red_wine$excellent, size=1599, replace=T)
bootstrap_p <- mean(p.bootstrap)
bootstrap_p.set[k] <- bootstrap_p
}

sd(bootstrap_p.set)
## [1] 0.008509628
conf.q.p <- quantile(bootstrap_p.set, probs=c(0.025, 0.975))
conf.q.p
##      2.5%     97.5% 
## 0.1188243 0.1525954
hist(bootstrap_p.set, freq = FALSE)
lines(density(bootstrap_p.set), lwd=5, col='blue')

#### c. Compare the two intervals. Is there any difference worth our attention? ##### There is no significant difference between the intervals calculated