Question

complete parts (a) through (c) below.
a. obtain the mean and median of the data.
the mean is 15.
(type an integer or a decimal. do not round.)
the median is 15.
(type an integer or a decimal. do not round.)
b. replace the 19 in the data set by 109 and again compute the mean and median. decide which measure of center works better here, and explain your answer.
the mean is 25.
(type an integer or a decimal. do not round.)
the median is 15.
(type an integer or a decimal. do not round.)
which center of measure works better here?
a. neither measure of center works for this data set. neither measure of center is typical of most of the data.
b. the mean works better here since it is more typical of most of the data.
c. the median works better here since it is more typical of most of the data.
d. both centers of measure work equally well here. they are both typical of most of the data.
c. for the data set in part (b), the mean is neither central nor typical for the data. the lack of what property of the mean accounts for this result?
a. the mean is not resistant to outliers. since the 19 in the data set was replaced by 109, the mean is pulled in that direction due to that single observation, and is therefore neither central nor typical for the data.
b. the mean has a lower probability of being observed. whereas observations in the data set have been observed and are likely to be obtained in repeated sampling, the mean is not necessarily observable, and is therefore neither central nor typical for the data.
c. the mean is not constant. since the mean changes depending on the observations in the data set and/or the number of observations, the mean cannot be representative of the underlying population.

Explanation:

Step1: Recall mean and median concepts

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, and the median is the middle - value when data is ordered.

Step2: Analyze part (b)

When 19 is replaced by 109, the mean changes from 15 to 25 because the mean is calculated by summing all values and dividing by the number of values. A large out - lier (109) affects the sum and thus the mean. The median remains 15 as it is the middle value and is not affected by extreme values. So, the median is a better measure of center as it is more typical of most of the data.

Step3: Analyze part (c)

The mean is not resistant to outliers. An outlier can pull the mean in its direction. In part (b), replacing 19 with 109 (an outlier) caused the mean to change significantly, making it neither central nor typical for the data.

Answer:

b. C. The median works better here since it is more typical of most of the data.
c. A. The mean is not resistant to outliers. Since the 19 in the data set was replaced by 109, the mean is pulled in that direction due to that single observation, and is therefore neither central nor typical for the data.