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.
246 261 267 272 276 278 281 283 284 285
287 287 289 290 293 295 295 298 308 505

(type an integer or decimal rounded to one decimal place as needed.)
the untrimmed mean is 294 pounds.
(type an integer or decimal rounded to one decimal place as needed.)
the 10% trimmed mean is 285.0 pounds.
(type an integer or decimal rounded to one decimal place as needed.)
the 20% trimmed mean is 285.7 pounds.
compare the values. choose the correct answer below.
a. all of the values are close to each other.
b. the untrimmed mean, 10% trimmed mean, and 20% trimmed mean are close to each other. however, the median is significantly different from those values.
c. the median, 10% trimmed mean, and 20% trimmed mean are close to each other. however, the untrimmed mean is significantly different from those values.
d. the median, untrimmed mean, and 20% trimmed mean are close to each other. however, the 10% trimmed mean is significantly different from those values.
e. the median, untrimmed mean, and 10% trimmed mean are close to each other. however, the 20% trimmed mean is significantly different from those values.

Explanation:

1. First, find the median. The data set has 20 values (count the numbers: 246, 261, 267, 272, 276, 278, 281, 283, 284, 285, 287, 287, 289, 290, 293, 295, 295, 298, 308, 505). The median for an even - numbered data set is the average of the 10th and 11th values. The 10th value is 285 and the 11th value is 287. So the median is $\frac{285 + 287}{2}=286$.
2. The untrimmed mean is 294, the 10% trimmed mean is 285.0, the 20% trimmed mean is 285.7, and the median is 286.

• The median (286), 10% trimmed mean (285.0), and 20% trimmed mean (285.7) are relatively close to each other.
• The untrimmed mean (294) is significantly different from these values because the untrimmed mean is affected by the extreme value 505 (a high outlier).

Answer:

C. The median, 10% trimmed mean, and 20% trimmed mean are close to each other. However, the untrimmed mean is significantly different from those values.