Solved on Jan 17, 2024

Monthly rents of 10 people change when one person's rent decreases from 1630to1630 to 1230. How does the median and mean change?

STEP 1

Assumptions
1. There are 10 people paying monthly rents.
2. The rents are already ordered from least to greatest.
3. The original rents are: 910,940,980,1000,1005,1020,1030,1045,1080,1630910, 940, 980, 1000, 1005, 1020, 1030, 1045, 1080, 1630.
4. One person's rent changes from 16301630 to 12301230.
5. We need to determine the change in the median and mean due to this change in rent.

STEP 2

First, we need to find the original median of the rents. Since there are 10 data points, the median will be the average of the 5th and 6th values.
Medianoriginal=5thvalue+6thvalue2Median_{original} = \frac{5th\, value + 6th\, value}{2}

STEP 3

Now, plug in the 5th and 6th values to calculate the original median.
Medianoriginal=1005+10202Median_{original} = \frac{1005 + 1020}{2}

STEP 4

Calculate the original median.
Medianoriginal=20252=1012.5Median_{original} = \frac{2025}{2} = 1012.5

STEP 5

Next, we need to find the new median after the rent change. The new rents will be:
910,940,980,1000,1005,1020,1030,1045,1080,1230910, 940, 980, 1000, 1005, 1020, 1030, 1045, 1080, 1230

STEP 6

Again, since there are 10 data points, the median will be the average of the 5th and 6th values.
Mediannew=5thvalue+6thvalue2Median_{new} = \frac{5th\, value + 6th\, value}{2}

STEP 7

Now, plug in the new 5th and 6th values to calculate the new median.
Mediannew=1005+10202Median_{new} = \frac{1005 + 1020}{2}

STEP 8

Calculate the new median.
Mediannew=20252=1012.5Median_{new} = \frac{2025}{2} = 1012.5

STEP 9

Compare the original median and the new median to determine the change.
Changeinmedian=MediannewMedianoriginalChange\, in\, median = Median_{new} - Median_{original}

STEP 10

Calculate the change in median.
Changeinmedian=1012.51012.5=0Change\, in\, median = 1012.5 - 1012.5 = 0

STEP 11

Now, we need to find the original mean of the rents. The mean is the sum of all values divided by the number of values.
Meanoriginal=SumofalloriginalrentsNumberofrentsMean_{original} = \frac{Sum\, of\, all\, original\, rents}{Number\, of\, rents}

STEP 12

Calculate the sum of all original rents.
Sumofalloriginalrents=910+940+980+1000+1005+1020+1030+1045+1080+1630Sum\, of\, all\, original\, rents = 910 + 940 + 980 + 1000 + 1005 + 1020 + 1030 + 1045 + 1080 + 1630

STEP 13

Calculate the sum.
Sumofalloriginalrents=10640Sum\, of\, all\, original\, rents = 10640

STEP 14

Now, calculate the original mean.
Meanoriginal=1064010Mean_{original} = \frac{10640}{10}

STEP 15

Calculate the original mean.
Meanoriginal=1064Mean_{original} = 1064

STEP 16

Next, we need to find the new mean after the rent change. The new sum of rents will be the original sum minus the old rent plus the new rent.
Sumofallnewrents=SumofalloriginalrentsOldrent+NewrentSum\, of\, all\, new\, rents = Sum\, of\, all\, original\, rents - Old\, rent + New\, rent

STEP 17

Now, plug in the values to calculate the new sum of rents.
Sumofallnewrents=106401630+1230Sum\, of\, all\, new\, rents = 10640 - 1630 + 1230

STEP 18

Calculate the new sum.
Sumofallnewrents=106401630+1230=10240Sum\, of\, all\, new\, rents = 10640 - 1630 + 1230 = 10240

STEP 19

Now, calculate the new mean.
Meannew=SumofallnewrentsNumberofrentsMean_{new} = \frac{Sum\, of\, all\, new\, rents}{Number\, of\, rents}

STEP 20

Calculate the new mean.
Meannew=1024010Mean_{new} = \frac{10240}{10}

STEP 21

Calculate the new mean.
Meannew=1024Mean_{new} = 1024

STEP 22

Compare the original mean and the new mean to determine the change.
Changeinmean=MeannewMeanoriginalChange\, in\, mean = Mean_{new} - Mean_{original}

STEP 23

Calculate the change in mean.
Changeinmean=10241064Change\, in\, mean = 1024 - 1064

STEP 24

Calculate the change in mean.
Changeinmean=40Change\, in\, mean = -40

STEP 25

Now we can answer the questions:
(a) What happens to the median? It stays the same.
(b) What happens to the mean? It decreases by $40.

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