Question: The monthly rents (in dollars) paid by 10 people are given below.
(Note that these are already ordered from least to greatest.)
830 , 920 , 950 , 975 , 1005 , 1020 , 1030 , 1080 , 1175 , 1825
Suppose that one of the people moves. His rent changes from $ 1825 to $ 1325 .
Answer the following.
(a) What happens to the ?
It decreases by
$ empty input box .
It increases by
$ empty input box .
It stays the same.
(b) What happens to the ?
It decreases by
$ empty input box .
It increases by
$ empty input box .
It stays the same.
1. **State the problem:** We have 10 monthly rents ordered from least to greatest: $$830, 920, 950, 975, 1005, 1020, 1030, 1080, 1175, 1825$$. One rent changes from $$1825$$ to $$1325$$. We want to find what happens to the median and the mean.
2. **Find the original median:** Since there are 10 values (even number), the median is the average of the 5th and 6th values.
$$\text{Median} = \frac{1005 + 1020}{2} = \frac{2025}{2} = 1012.5$$
3. **Find the new list after change:** Replace $$1825$$ with $$1325$$ and reorder:
$$830, 920, 950, 975, 1005, 1020, 1030, 1080, 1175, 1325$$
4. **Find the new median:** Again average of 5th and 6th values:
$$\text{Median} = \frac{1005 + 1020}{2} = 1012.5$$
5. **Conclusion for median:** The median stays the same.
6. **Find the original mean:** Sum all original rents:
$$830 + 920 + 950 + 975 + 1005 + 1020 + 1030 + 1080 + 1175 + 1825 = 10810$$
Mean:
$$\frac{10810}{10} = 1081$$
7. **Find the new mean:** Replace $$1825$$ with $$1325$$, sum becomes:
$$10810 - 1825 + 1325 = 10310$$
New mean:
$$\frac{10310}{10} = 1031$$
8. **Calculate change in mean:**
$$1081 - 1031 = 50$$ decrease.
9. **Final answers:**
(a) Median stays the same.
(b) Mean decreases by $$50$$.