Let's address each part of the question step-by-step.
Step 1: Original Calculation
1.1 Calculate the original mean (average) rent:
Given the rents are:
[tex]\[ 870, 880, 930, 955, 980, 1020, 1070, 1105, 1180, 1380 \][/tex]
To find the mean, sum all the rents and divide by the number of rents:
[tex]\[ \text{Mean} = \frac{870 + 880 + 930 + 955 + 980 + 1020 + 1070 + 1105 + 1180 + 1380}{10} \][/tex]
So,
[tex]\[ \text{Mean} = \frac{10370}{10} = 1037 \][/tex]
1.2 Calculate the original median rent:
Since there are 10 rents, the median will be the average of the 5th and 6th values in the ordered list:
Median is the average of 980 and 1020:
[tex]\[ \text{Median} = \frac{980 + 1020}{2} = 1000 \][/tex]
Step 2: Change in Rent
2.1 Modify the rent from [tex]$1380 to $[/tex]1730.
The new list of rents is:
[tex]\[ 870, 880, 930, 955, 980, 1020, 1070, 1105, 1180, 1730 \][/tex]
2.2 Calculate the new mean:
Sum the new rents and divide by 10:
[tex]\[ \text{New Sum} = 870 + 880 + 930 + 955 + 980 + 1020 + 1070 + 1105 + 1180 + 1730 \][/tex]
[tex]\[ \text{New Sum} = 10720 \][/tex]
[tex]\[ \text{New Mean} = \frac{10720}{10} = 1072 \][/tex]
Step 3: Analyze Changes
3.1 Change in mean:
New mean [tex]\( 1072 \)[/tex] compared to original mean [tex]\( 1037 \)[/tex]:
[tex]\[ \text{Mean Change} = 1072 - 1037 = 35 \][/tex]
Thus, the mean increases by [tex]\( \$35 \)[/tex].
3.2 Change in median:
The middle two values (980 and 1020) remain unchanged when recalculating the median after replacing [tex]$1380 with $[/tex]1730 because the new value didn't affect the middle position.
So, the median remains:
[tex]\[ \text{Median} = 1000 \][/tex]
Final Answer:
[tex]\[
\begin{tabular}{|l|l|}
\hline
(a) What happens to the mean? & It decreases by \$ \square. \\
& It increases by \$ 35. \\
& It stays the same. \\
\hline
(b) What happens to the median? & It decreases by \$ \square. \\
& It increases by \$ \square. \\
& It stays the same. \\
\hline
\end{tabular}
\][/tex]
