Question

Because the mean is very sensitive to extreme​ values, it is not a resistant measure of center. By deleting some low values and high​ values, the trimmed mean is more resistant. To find the​ 10% trimmed mean for a data​ set, first arrange the data in​ order, then delete the bottom​ 10% of the values and delete the top​ 10% of the​ values, then calculate the mean of the remaining values. Use the axial loads​ (pounds) of aluminum cans listed below for cans that are 0.0111 in. thick. Identify any​ outliers, then compare the​ median, mean,​ 10% trimmed​ mean, and​ 20% trimmed mean. 247 261 267 274 275 278 281 284 284 285 286 287 289 292 294 295 297 300 310 504

Answer 1

Here is given below R-code and output with explanation

> x<- c(247, 261, 267, 274, 275, 278, 281, 284, 284, 285, 286, 287, 289, 292, 294, 295, 297, 300, 310, 504) # given Data set
> length(x)
[1] 20
> mean(x)
[1] 294.5

Calculations for outlier:-

Q1= 5th observation =275
Q3= 15th observation =294
IQR=(Q3-Q1)/2=9.5
Outlier Range=Q1-(1.5)*IQR and Q3+(1.5)*IQR i.e. (260.75, 308.25)

Therefore outlier data i,e outside the above range are (310 and 504)

Now after elimination of outlier data the cleaned data set is
(247, 261, 267, 274, 275, 278, 281, 284, 284, 285, 286, 287, 289, 292, 294, 295, 297, 300)
> x_clean<-c(247, 261, 267, 274, 275, 278, 281, 284, 284, 285, 286, 287, 289, 292, 294, 295, 297, 300)
> mean(x_clean)
[1] 282
> median(x_clean)
[1] 284.5

Data set after 10% trimmed from both end we have the following
> x_10T<- c(267, 274, 275, 278, 281, 284, 284, 285, 286, 287, 289, 292, 294, 295, 297, 300)
> length(x_10T)
[1] 16
> mean(x_10T)
[1] 285.5 # mean of 10% trimmed data

> median(x_10T)
[1] 285.5

Data set after 20% trimed from both end we have
> x_20T<- c(275, 278, 281, 284, 284, 285, 286, 287, 289, 292, 294, 295)
> mean(x_20T)
[1] 285.8333 # mean of 20% trimmed data
> median(x_20T)
[1] 285.5

Therefore it is clear that (a) mean of 10% trimmed data= 285.5 <mean of 20% trimmed data= 285.8333

(b) median of 10% trimmed data= 285.5 and median of 20% trimmed data= 285.5 are. same.