Question

data set f consists of 55 integers between 170 and 290. data set g consists of all the integers in data set f as well as the integer 10. which of the following must be less for data set f than for data set g? i. the mean ii. the median a) i only b) ii only c) i and ii d) neither i nor ii

Explanation:

Step1: Recall mean formula

The mean of a data - set $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n$ is the number of data - points and $\sum_{i = 1}^{n}x_{i}$ is the sum of the data - points. Let the sum of the 55 integers in data - set $F$ be $S$. The mean of data - set $F$, $\bar{x}_F=\frac{S}{55}$. The sum of the data - points in data - set $G$ is $S + 10$, and the number of data - points in data - set $G$ is $n_G=56$. The mean of data - set $G$, $\bar{x}_G=\frac{S + 10}{56}$.
We know that $\bar{x}_F-\bar{x}_G=\frac{S}{55}-\frac{S + 10}{56}=\frac{56S-55(S + 10)}{55\times56}=\frac{56S-55S-550}{55\times56}=\frac{S - 550}{55\times56}$. Since the integers in data - set $F$ are between 170 and 290, $S>55\times170 = 9350$. So, $\bar{x}_F>\bar{x}_G$.

Step2: Recall median concept

Data - set $F$ has 55 (an odd number) data - points. The median of data - set $F$ is the 28th ordered data - point. Data - set $G$ has 56 (an even number) data - points. The median of data - set $G$ is the average of the 28th and 29th ordered data - points. Since 10 is the smallest value in data - set $G$, when we order the data - set $G$, the 28th and 29th ordered data - points in $G$ are the same as the 28th and 29th ordered data - points in $F$ (because 10 is the smallest value and will be at the beginning of the ordered list of $G$). So the median of $F$ is equal to the median of $G$.

Answer:

A. I only