Let's analyze the given situation step-by-step to determine how the changes in rent affect the mean and median.
### Step (a): Effect on the Mean
1. Original Data: The monthly rents paid by the 9 people are:
[tex]\[
835, 930, 965, 990, 1000, 1015, 1075, 1105, 1130
\][/tex]
2. Calculating Original Mean:
To find the mean, sum up all the rents and divide by the number of people.
[tex]\[
\text{Sum of rents} = 835 + 930 + 965 + 990 + 1000 + 1015 + 1075 + 1105 + 1130 = 9045
\][/tex]
[tex]\[
\text{Number of people} = 9
\][/tex]
[tex]\[
\text{Original mean} = \frac{9045}{9} = 1005.0
\][/tex]
3. New Data: One person's rent changes from \[tex]$1130 to \$[/tex]1040. The new rents are:
[tex]\[
835, 930, 965, 990, 1000, 1015, 1075, 1105, 1040
\][/tex]
4. Calculating New Mean:
Adjust the sum of the rents by subtracting the old rent and adding the new rent:
[tex]\[
\text{New sum of rents} = 9045 - 1130 + 1040 = 8955
\][/tex]
[tex]\[
\text{New mean} = \frac{8955}{9} = 995.0
\][/tex]
5. Change in Mean:
The mean decreases from 1005.0 to 995.0.
[tex]\[
\text{Change in mean} = 1005.0 - 995.0 = 10.0
\][/tex]
So, the mean decreases by \[tex]$10.0.
### Step (b): Effect on the Median
1. Original Data: The ordered rents are the same as given initially.
\[
835, 930, 965, 990, 1000, 1015, 1075, 1105, 1130
\]
2. Calculating Original Median:
For an odd number of observations, the median is the middle number.
The 5th rent (middle of 9) is:
\[
\text{Original median} = 1000
\]
3. New Data: The new ordered rents after one person changes their rent:
\[
835, 930, 965, 990, 1000, 1015, 1075, 1105, 1040
\]
Sorting the new rents:
\[
835, 930, 965, 990, 1000, 1015, 1040, 1075, 1105
\]
4. Calculating New Median:
Again, for an odd number of observations, the median is the middle number. The 5th rent (middle of 9) remains:
\[
\text{New median} = 1000
\]
5. Change in Median:
The median does not change.
\[
\text{Change in median} = 1000 - 1000 = 0
\]
So, the median stays the same.
### Summary
- Effect on Mean: The mean decreases by \$[/tex]10.0.
- Effect on Median: The median stays the same.
Thus, filling out the table:
\begin{tabular}{|l|l|}
\hline (a) What happens to the mean? & O It decreases by [tex]$\$[/tex] 10.0[tex]$. \\
& It increases by $[/tex]\[tex]$ \square$[/tex]. \\
& It stays the same. \\
\hline (b) What happens to the median? & O It stays the same. \\
& It decreases by [tex]$\$[/tex] \square[tex]$. \\
& It increases by $[/tex]\[tex]$ \square$[/tex]. \\
\hline
\end{tabular}
