To analyze how changing one rent value affects the median and mean of the rents, let's break it down step-by-step.
### Initial Rents
The original rents are:
[tex]\[ 820, 875, 920, 935, 1000, 1055, 1085, 1150 \][/tex]
### Initial Median
The median is the middle value in an ordered list. For an even number of data points, the median is the average of the two middle numbers:
[tex]\[ \text{Median} = \frac{935 + 1000}{2} = 967.5 \][/tex]
### Initial Mean
The mean is calculated by adding all the values and dividing by the number of values:
[tex]\[ \text{Mean} = \frac{820 + 875 + 920 + 935 + 1000 + 1055 + 1085 + 1150}{8} = \frac{7840}{8} = 980 \][/tex]
### Updated Rents
One of the rents changes from \[tex]$1150 to \$[/tex]1030. So, the new rents are:
[tex]\[ 820, 875, 920, 935, 1000, 1055, 1085, 1030 \][/tex]
We need to reorder the rents to find the new median:
[tex]\[ 820, 875, 920, 935, 1000, 1030, 1055, 1085 \][/tex]
### New Median
With the updated list, the median is still the average of the middle two numbers (935 and 1000):
[tex]\[ \text{New Median} = \frac{935 + 1000}{2} = 967.5 \][/tex]
We see that the median stays the same.
### New Mean
To find the new mean, we add the updated rent values and divide by the same number of rents:
[tex]\[ \text{New Mean} = \frac{820 + 875 + 920 + 935 + 1000 + 1030 + 1055 + 1085}{8} = \frac{7720}{8} = 965 \][/tex]
The mean decreases.
### Differences
- The difference in the median is:
[tex]\[ \text{Difference in Median} = 967.5 - 967.5 = 0 \][/tex]
- The difference in the mean is:
[tex]\[ \text{Difference in Mean} = 980 - 965 = 15 \][/tex]
Given the above calculations, we conclude the following:
## Answers
\begin{tabular}{|c|c|c|}
\hline (a) & What happens to the median? & \begin{tabular}{l}
It stays the same.
\end{tabular} \\
\hline (b) & What happens to the mean? & \begin{tabular}{l}
It decreases by [tex]$\$[/tex] 15$. \\
\end{tabular} \\
\hline
\end{tabular}
