consider the data set 11,12,13,14,15,16,17,18,19. complete parts (a) through (c) below.\na. obtain the mean…

consider the data set 11,12,13,14,15,16,17,18,19. complete parts (a) through (c) below.\na. obtain the mean and median of the data.\nthe mean is 15.\n(type an integer or a decimal. do not round.)\nthe median is 15.\n(type an integer or a decimal. do not round.)\nb. 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.\nthe mean is $\frac{224}{9}$.\n(type an integer or a decimal. do not round.)\nthe median is 15.\n(type an integer or a decimal. do not round.)

consider the data set 11,12,13,14,15,16,17,18,19. complete parts (a) through (c) below.\na. obtain the mean and median of the data.\nthe mean is 15.\n(type an integer or a decimal. do not round.)\nthe median is 15.\n(type an integer or a decimal. do not round.)\nb. 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.\nthe mean is $\frac{224}{9}$.\n(type an integer or a decimal. do not round.)\nthe median is 15.\n(type an integer or a decimal. do not round.)

Answer

Explanation:

Step1: Calculate original mean

The formula for the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. Here $n = 9$, and $\sum_{i=1}^{9}x_{i}=11 + 12+13+14+15+16+17+18+19=\frac{(11 + 19)\times9}{2}=135$. So $\bar{x}=\frac{135}{9}=15$.

Step2: Find original median

The data set $11,12,13,14,15,16,17,18,19$ is already in ascending - order. Since $n = 9$ (odd), the median is the $\left(\frac{n + 1}{2}\right)$-th value. $\frac{9+1}{2}=5$-th value, which is $15$.

Step3: Calculate new mean

Replace 19 with 109. Then $\sum_{i=1}^{9}x_{i}=11 + 12+13+14+15+16+17+18+109=135-19 + 109=224$. So the new mean $\bar{x}=\frac{224}{9}$.

Step4: Find new median

The new data set in ascending - order is $11,12,13,14,15,16,17,18,109$. Since $n = 9$ (odd), the median is the 5 - th value, which is $15$.

Step5: Decide better measure of center

The median works better here. The mean is affected by the out - lier 109 (it has increased significantly from 15 to $\frac{224}{9}\approx24.89$), while the median remains the same. The median is more resistant to extreme values.

Answer:

a. Mean: 15, Median: 15 b. Mean: $\frac{224}{9}$, Median: 15. The median works better because it is more resistant to extreme values.