Adams HW6
Jenea Adams
December 9, 2021
12.74
dan = read.csv("EX12-74DANDRUFF.csv")
dan## OBS Treatment Flaking
## 1 1 PyrI 17
## 2 2 PyrI 16
## 3 3 PyrI 18
## 4 4 PyrI 17
## 5 5 PyrI 18
## 6 6 PyrI 16
## 7 7 PyrI 17
## 8 8 PyrI 20
## 9 9 PyrI 17
## 10 10 PyrI 17
## 11 11 PyrI 15
## 12 12 PyrI 18
## 13 13 PyrI 17
## 14 14 PyrI 17
## 15 15 PyrI 17
## 16 16 PyrI 15
## 17 17 PyrI 17
## 18 18 PyrI 16
## 19 19 PyrI 20
## 20 20 PyrI 17
## 21 21 PyrI 17
## 22 22 PyrI 17
## 23 23 PyrI 18
## 24 24 PyrI 18
## 25 25 PyrI 17
## 26 26 PyrI 18
## 27 27 PyrI 18
## 28 28 PyrI 19
## 29 29 PyrI 16
## 30 30 PyrI 17
## 31 31 PyrI 18
## 32 32 PyrI 20
## 33 33 PyrI 19
## 34 34 PyrI 18
## 35 35 PyrI 19
## 36 36 PyrI 17
## 37 37 PyrI 19
## 38 38 PyrI 15
## 39 39 PyrI 19
## 40 40 PyrI 18
## 41 41 PyrI 19
## 42 42 PyrI 16
## 43 43 PyrI 19
## 44 44 PyrI 17
## 45 45 PyrI 18
## 46 46 PyrI 17
## 47 47 PyrI 16
## 48 48 PyrI 16
## 49 49 PyrI 17
## 50 50 PyrI 18
## 51 51 PyrI 17
## 52 52 PyrI 17
## 53 53 PyrI 15
## 54 54 PyrI 18
## 55 55 PyrI 19
## 56 56 PyrI 17
## 57 57 PyrI 18
## 58 58 PyrI 18
## 59 59 PyrI 18
## 60 60 PyrI 18
## 61 61 PyrI 18
## 62 62 PyrI 20
## 63 63 PyrI 16
## 64 64 PyrI 18
## 65 65 PyrI 19
## 66 66 PyrI 15
## 67 67 PyrI 18
## 68 68 PyrI 16
## 69 69 PyrI 16
## 70 70 PyrI 17
## 71 71 PyrI 19
## 72 72 PyrI 15
## 73 73 PyrI 17
## 74 74 PyrI 18
## 75 75 PyrI 17
## 76 76 PyrI 18
## 77 77 PyrI 18
## 78 78 PyrI 17
## 79 79 PyrI 18
## 80 80 PyrI 16
## 81 81 PyrI 17
## 82 82 PyrI 17
## 83 83 PyrI 17
## 84 84 PyrI 17
## 85 85 PyrI 17
## 86 86 PyrI 17
## 87 87 PyrI 17
## 88 88 PyrI 18
## 89 89 PyrI 18
## 90 90 PyrI 17
## 91 91 PyrI 17
## 92 92 PyrI 18
## 93 93 PyrI 18
## 94 94 PyrI 18
## 95 95 PyrI 18
## 96 96 PyrI 17
## 97 97 PyrI 17
## 98 98 PyrI 17
## 99 99 PyrI 17
## 100 100 PyrI 18
## 101 101 PyrI 16
## 102 102 PyrI 19
## 103 103 PyrI 19
## 104 104 PyrI 17
## 105 105 PyrI 19
## 106 106 PyrI 17
## 107 107 PyrI 15
## 108 108 PyrI 16
## 109 109 PyrI 18
## 110 110 PyrI 18
## 111 111 PyrI 17
## 112 112 PyrI 17
## 113 113 PyrII 17
## 114 114 PyrII 17
## 115 115 PyrII 18
## 116 116 PyrII 19
## 117 117 PyrII 16
## 118 118 PyrII 18
## 119 119 PyrII 18
## 120 120 PyrII 16
## 121 121 PyrII 18
## 122 122 PyrII 19
## 123 123 PyrII 19
## 124 124 PyrII 17
## 125 125 PyrII 19
## 126 126 PyrII 18
## 127 127 PyrII 17
## 128 128 PyrII 18
## 129 129 PyrII 16
## 130 130 PyrII 19
## 131 131 PyrII 16
## 132 132 PyrII 16
## 133 133 PyrII 19
## 134 134 PyrII 17
## 135 135 PyrII 19
## 136 136 PyrII 14
## 137 137 PyrII 17
## 138 138 PyrII 18
## 139 139 PyrII 17
## 140 140 PyrII 19
## 141 141 PyrII 18
## 142 142 PyrII 16
## 143 143 PyrII 16
## 144 144 PyrII 15
## 145 145 PyrII 16
## 146 146 PyrII 18
## 147 147 PyrII 16
## 148 148 PyrII 14
## 149 149 PyrII 18
## 150 150 PyrII 17
## 151 151 PyrII 17
## 152 152 PyrII 17
## 153 153 PyrII 16
## 154 154 PyrII 18
## 155 155 PyrII 19
## 156 156 PyrII 16
## 157 157 PyrII 16
## 158 158 PyrII 16
## 159 159 PyrII 17
## 160 160 PyrII 17
## 161 161 PyrII 19
## 162 162 PyrII 18
## 163 163 PyrII 19
## 164 164 PyrII 18
## 165 165 PyrII 18
## 166 166 PyrII 17
## 167 167 PyrII 16
## 168 168 PyrII 18
## 169 169 PyrII 19
## 170 170 PyrII 14
## 171 171 PyrII 16
## 172 172 PyrII 18
## 173 173 PyrII 16
## 174 174 PyrII 16
## 175 175 PyrII 15
## 176 176 PyrII 19
## 177 177 PyrII 17
## 178 178 PyrII 19
## 179 179 PyrII 17
## 180 180 PyrII 20
## 181 181 PyrII 16
## 182 182 PyrII 16
## 183 183 PyrII 17
## 184 184 PyrII 16
## 185 185 PyrII 16
## 186 186 PyrII 16
## 187 187 PyrII 16
## 188 188 PyrII 16
## 189 189 PyrII 18
## 190 190 PyrII 19
## 191 191 PyrII 17
## 192 192 PyrII 17
## 193 193 PyrII 15
## 194 194 PyrII 17
## 195 195 PyrII 19
## 196 196 PyrII 17
## 197 197 PyrII 20
## 198 198 PyrII 17
## 199 199 PyrII 19
## 200 200 PyrII 15
## 201 201 PyrII 17
## 202 202 PyrII 18
## 203 203 PyrII 16
## 204 204 PyrII 17
## 205 205 PyrII 18
## 206 206 PyrII 17
## 207 207 PyrII 16
## 208 208 PyrII 16
## 209 209 PyrII 17
## 210 210 PyrII 18
## 211 211 PyrII 18
## 212 212 PyrII 18
## 213 213 PyrII 16
## 214 214 PyrII 16
## 215 215 PyrII 18
## 216 216 PyrII 17
## 217 217 PyrII 21
## 218 218 PyrII 17
## 219 219 PyrII 19
## 220 220 PyrII 16
## 221 221 PyrII 17
## 222 222 Keto 17
## 223 223 Keto 17
## 224 224 Keto 17
## 225 225 Keto 15
## 226 226 Keto 14
## 227 227 Keto 16
## 228 228 Keto 16
## 229 229 Keto 17
## 230 230 Keto 15
## 231 231 Keto 15
## 232 232 Keto 16
## 233 233 Keto 17
## 234 234 Keto 17
## 235 235 Keto 16
## 236 236 Keto 16
## 237 237 Keto 15
## 238 238 Keto 16
## 239 239 Keto 17
## 240 240 Keto 16
## 241 241 Keto 15
## 242 242 Keto 18
## 243 243 Keto 17
## 244 244 Keto 16
## 245 245 Keto 17
## 246 246 Keto 15
## 247 247 Keto 15
## 248 248 Keto 16
## 249 249 Keto 15
## 250 250 Keto 18
## 251 251 Keto 16
## 252 252 Keto 16
## 253 253 Keto 18
## 254 254 Keto 16
## 255 255 Keto 16
## 256 256 Keto 17
## 257 257 Keto 16
## 258 258 Keto 15
## 259 259 Keto 15
## 260 260 Keto 16
## 261 261 Keto 17
## 262 262 Keto 16
## 263 263 Keto 18
## 264 264 Keto 18
## 265 265 Keto 16
## 266 266 Keto 17
## 267 267 Keto 15
## 268 268 Keto 15
## 269 269 Keto 16
## 270 270 Keto 16
## 271 271 Keto 14
## 272 272 Keto 15
## 273 273 Keto 17
## 274 274 Keto 16
## 275 275 Keto 15
## 276 276 Keto 17
## 277 277 Keto 16
## 278 278 Keto 16
## 279 279 Keto 15
## 280 280 Keto 16
## 281 281 Keto 14
## 282 282 Keto 17
## 283 283 Keto 16
## 284 284 Keto 16
## 285 285 Keto 15
## 286 286 Keto 16
## 287 287 Keto 16
## 288 288 Keto 16
## 289 289 Keto 16
## 290 290 Keto 17
## 291 291 Keto 15
## 292 292 Keto 16
## 293 293 Keto 16
## 294 294 Keto 16
## 295 295 Keto 15
## 296 296 Keto 17
## 297 297 Keto 16
## 298 298 Keto 17
## 299 299 Keto 16
## 300 300 Keto 16
## 301 301 Keto 16
## 302 302 Keto 16
## 303 303 Keto 18
## 304 304 Keto 15
## 305 305 Keto 17
## 306 306 Keto 16
## 307 307 Keto 15
## 308 308 Keto 15
## 309 309 Keto 15
## 310 310 Keto 16
## 311 311 Keto 17
## 312 312 Keto 15
## 313 313 Keto 15
## 314 314 Keto 14
## 315 315 Keto 16
## 316 316 Keto 17
## 317 317 Keto 16
## 318 318 Keto 16
## 319 319 Keto 15
## 320 320 Keto 17
## 321 321 Keto 17
## 322 322 Keto 17
## 323 323 Keto 16
## 324 324 Keto 16
## 325 325 Keto 17
## 326 326 Keto 15
## 327 327 Keto 16
## 328 328 Placebo 30
## 329 329 Placebo 29
## 330 330 Placebo 31
## 331 331 Placebo 31
## 332 332 Placebo 29
## 333 333 Placebo 29
## 334 334 Placebo 28
## 335 335 Placebo 29
## 336 336 Placebo 26
## 337 337 Placebo 29
## 338 338 Placebo 29
## 339 339 Placebo 28
## 340 340 Placebo 31
## 341 341 Placebo 30
## 342 342 Placebo 28
## 343 343 Placebo 28
## 344 344 Placebo 30
## 345 345 Placebo 31
## 346 346 Placebo 31
## 347 347 Placebo 31
## 348 348 Placebo 28
## 349 349 Placebo 32
## 350 350 Placebo 32
## 351 351 Placebo 29
## 352 352 Placebo 28
## 353 353 Placebo 26
## 354 354 Placebo 29
## 355 355 Placebo 31head(dan)## OBS Treatment Flaking
## 1 1 PyrI 17
## 2 2 PyrI 16
## 3 3 PyrI 18
## 4 4 PyrI 17
## 5 5 PyrI 18
## 6 6 PyrI 16res.aov_dan <- aov(Flaking ~ Treatment, data = dan )
summary(res.aov_dan)## Df Sum Sq Mean Sq F value Pr(>F)
## Treatment 3 4151 1383.8 967.8 <2e-16 ***
## Residuals 351 502 1.4
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1pairwise.t.test(dan$Flaking, dan$Treatment, p.adj = "bonf")##
## Pairwise comparisons using t tests with pooled SD
##
## data: dan$Flaking and dan$Treatment
##
## Keto Placebo PyrI
## Placebo < 2e-16 - -
## PyrI 5.8e-15 < 2e-16 -
## PyrII 2.3e-11 < 2e-16 1
##
## P value adjustment method: bonferroniBonferroni correction leads to p-values that are all significant besides the comparison between the PyrII and PyrI groups, which is supported in the contrast analysis. This shows that overall, the placebo has a significantly different mean that the other non-placebo groups.
13.11
grocery = read.csv("EX13-11SMART2.csv")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, unionwith = filter(grocery, Smartcart == "With")
wo = filter(grocery, Smartcart == "Without")a)
xbar_with = mean(with$TotalCost)
xbar_wo = mean(wo$TotalCost)
barplot(c(xbar_with, xbar_wo), names.arg = c("With", "Without"),
col = c("#eb8060", "#b9e38d"))
The plot shows that the means of the two groups are quite close. Without is slightly higher than With real-time feedback.
b)
res.aov_grocery <- aov(TotalCost ~ Smartcart, data = grocery )
summary(res.aov_grocery)## Df Sum Sq Mean Sq F value Pr(>F)
## Smartcart 1 58 57.91 1.026 0.312
## Residuals 192 10838 56.45c)
The degrees of freedeomfor the residuls is 192, the F statistic if 1.026 and the p-value is 0.312.
Based on this analysis, the estimated effects may be unbalanced, but the p-value suggest that there is not enough evidence to conclude that the means are significantly different ebtween the two groups. Therefore, there is not enough evidence to suggest that receiving real-time feedback has a significant effect on the total cost of the shopping cart.
13.22
a)
biling = read.csv("EX13-22BILING.csv")young_mon = filter(biling, Age == "Young" & Ling =="Mono")
old_mon = filter(biling, Age == "Old" & Ling =="Mono")
young_bi = filter(biling, Age == "Young" & Ling =="Bi")
old_bi = filter(biling, Age == "Old" & Ling =="Bi")xbar_young_mon = mean(young_mon$Time)
xbar_old_mon = mean(old_mon$Time)
xbar_young_bi = mean(young_bi$Time)
xbar_old_bi = mean(old_bi$Time)sd_young_mon = sd(young_mon$Time)
sd_old_mon = sd(old_mon$Time)
sd_young_bi = sd(young_bi$Time)
sd_old_bi = sd(old_bi$Time)sample_sizes_biling = c(20, 20, 20, 20)
means_biling = c(xbar_young_mon, xbar_old_mon, xbar_young_bi, xbar_old_bi)
sds_biling = c(sd_young_mon, sd_old_mon, sd_young_bi, sd_old_bi)
data.frame(sample_sizes_biling, means_biling, sds_biling)## sample_sizes_biling means_biling sds_biling
## 1 20 820.70 47.38543
## 2 20 996.85 53.59524
## 3 20 785.65 42.04293
## 4 20 919.30 54.38663sqrt(54.38)/sqrt(42.02)## [1] 1.137605The rule for examining standard deviations in a pooled states that if the largest standard deviation is less than twice the smallest standard deviation, we can use methods based on the assumption of equal standard deviations, and our results will still be approximately correct. In this case, the ratio of the largest to smallest standard deviation is \(\frac{\sqrt{54.38}}{\sqrt{42.04}} = 1.137 < 2\); therefore, we can pool the standard deviations as it meets the assumptions.
b)
par(mfrow = c(2,2))
hist(young_mon$Time, breaks = 25)
hist(old_mon$Time, breaks = 25)
hist(young_bi$Time, breaks = 25)
hist(old_bi$Time, breaks = 25)
It appears that young groups are approximately normal, but the old groups are skewed or widely dispersed. Therefore, I would not make the qualitative conclusion that the old groups are approximately Normal.
13.23
a)
interaction.plot(x.factor = biling$Age, #x-axis variable
trace.factor = biling$Ling, #variable for lines
response = biling$Time, #y-axis variable
fun = median, #metric to plot
ylab = "Time",
xlab = "Language Level",
col = c("pink", "blue"),
lty = 1, #line type
lwd = 2, #line width
trace.label = "Languages Known")
We might find an interaction between age and lingualism because the more we age, the hard it becomes to learn a language and the more brain function declines in general.
b)
res.aov_biling <- aov(Time ~ Age + Ling, data= biling )
summary(res.aov_biling)## Df Sum Sq Mean Sq F value Pr(>F)
## Age 1 479880 479880 188.5 < 2e-16 ***
## Ling 1 63394 63394 24.9 3.66e-06 ***
## Residuals 77 196055 2546
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The F-value for Age and Linguistic ability is 188.5 and 24.9,r espectively. Both variables has a significant p-value of less than 0.001. Both variables have a degree of freedom of 1 and the residuls df = 77.
c)
This analysis suggests that an interaction effect between age and bilinguilism is abesnt since the p-values are significant for both. This suggests that both Age and Bilinguilism contribute to reaction times for the cognitive test.
16.18
library(dplyr)
drp_data = read.csv("EX16-18DRP.csv")
head(drp_data)## id group g drp
## 1 1 Treat 0 24
## 2 2 Treat 0 56
## 3 3 Treat 0 43
## 4 4 Treat 0 59
## 5 5 Treat 0 58
## 6 6 Treat 0 52drp_data <- drp_data %>%
mutate(group = as.factor(group)) a)
meandiff <- function(d, i){
d = d[i,]
y = tapply(d$drp, d$group, mean)
y[1]-y[2]
} library(boot)
drp_boot = boot(drp_data, meandiff, R = 2000, strata = drp_data$group)
drp_boot##
## STRATIFIED BOOTSTRAP
##
##
## Call:
## boot(data = drp_data, statistic = meandiff, R = 2000, strata = drp_data$group)
##
##
## Bootstrap Statistics :
## original bias std. error
## t1* -9.954451 -0.04919565 4.274403The bootstrap standard error is 4.159
b)
plot(drp_boot) #graphing bootstrap distribution
mean(drp_boot$t) #checking bootstrap mean## [1] -10.00365The bootstrap t confidence interval is appropriate because the bootstrap distribution is approximately Normal
boot.ci(boot.out = drp_boot, type = "norm") ## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
## Based on 2000 bootstrap replicates
##
## CALL :
## boot.ci(boot.out = drp_boot, type = "norm")
##
## Intervals :
## Level Normal
## 95% (-18.283, -1.528 )
## Calculations and Intervals on Original Scalec)
My bootstrap results are nearly identical to what is shown in example 7.14. this suggests that this confidence interval can be captured in multiple ways depending on the structure of the data. The actual mean of the boostrapoed data does in fact lie within this interval.
16.24
theta <- function(data,i) {
d = data[i]
mean(d)
}# tvtime = read.csv("EX16-24TVTIME.csv")
time = c(3, 16.5, 10.5, 40.5, 5.5, 33.5, 0, 6.5)
tvtime = data.frame(time) a)
xbar_tvtime = mean(tvtime$Time, na.rm = T)## Warning in mean.default(tvtime$Time, na.rm = T): argument is not numeric or
## logical: returning NAtvtime_boot = boot(data = tvtime$time, statistic = theta, R = 2000)plot(tvtime_boot)
Rhe plot show an approximately Normal distribution with a narrow variance about the mean. The observed xbar is 14.5 and there seems to be a tight grouping of means around that value for t*.
b)
boot.ci(boot.out = tvtime_boot, type = "norm")## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
## Based on 2000 bootstrap replicates
##
## CALL :
## boot.ci(boot.out = tvtime_boot, type = "norm")
##
## Intervals :
## Level Normal
## 95% ( 4.84, 24.38 )
## Calculations and Intervals on Original ScaleThe 95% bootstrap confidence interval for \(\mu\) is (5.11, 23.86).
c)
t.test(tvtime$time)##
## One Sample t-test
##
## data: tvtime$time
## t = 2.761, df = 7, p-value = 0.02806
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
## 2.081702 26.918298
## sample estimates:
## mean of x
## 14.5The usual t interval is (2.08, 26.91), which is wider than the one capture via bootstrapping of (5.11, 23.86). Neither interval includes 0. The wider interval for the t procedure could eb due to the small sample size and the fact that bootstrapping creates a new distribution from the original data directly.