Solved on Jan 17, 2024

Monthly rents of 10 people change when one person's rent decreases from $1630 to$ 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: $910, 940, 980, 1000, 1005, 1020, 1030, 1045, 1080, 1630$ .
4. One person's rent changes from $1630$ to $1230$ .
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.
$Median_{original} = \frac{5th\, value + 6th\, value}{2}$

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

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

STEP 7

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

STEP 8

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

STEP 9

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

STEP 10

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

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.
$Mean_{original} = \frac{Sum\, of\, all\, original\, rents}{Number\, of\, rents}$

STEP 12

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

STEP 13

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

STEP 14

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

STEP 15

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

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.
$Sum\, of\, all\, new\, rents = Sum\, of\, all\, original\, rents - Old\, rent + New\, rent$

STEP 17

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

STEP 18

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

STEP 19

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

STEP 20

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

STEP 21

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

STEP 22

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

STEP 23

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

STEP 24

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

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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Monthly rents of 10 people change when one person's rent decreases from 1630to1630 to 1630to1230. How does the median and mean change?

STEP 1

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=5th value+6th value2Median_{original} = \frac{5th\, value + 6th\, value}{2}Medianoriginal​=25thvalue+6thvalue​

STEP 3

Now, plug in the 5th and 6th values to calculate the original median.Medianoriginal=1005+10202Median_{original} = \frac{1005 + 1020}{2}Medianoriginal​=21005+1020​

STEP 4

Calculate the original median.Medianoriginal=20252=1012.5Median_{original} = \frac{2025}{2} = 1012.5Medianoriginal​=22025​=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, 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=5th value+6th value2Median_{new} = \frac{5th\, value + 6th\, value}{2}Mediannew​=25thvalue+6thvalue​

STEP 7

Now, plug in the new 5th and 6th values to calculate the new median.Mediannew=1005+10202Median_{new} = \frac{1005 + 1020}{2}Mediannew​=21005+1020​

STEP 8

Calculate the new median.Mediannew=20252=1012.5Median_{new} = \frac{2025}{2} = 1012.5Mediannew​=22025​=1012.5

STEP 9

Compare the original median and the new median to determine the change.Change in median=Mediannew−MedianoriginalChange\, in\, median = Median_{new} - Median_{original}Changeinmedian=Mediannew​−Medianoriginal​

STEP 10

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

STEP 11

STEP 12

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

STEP 13

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

STEP 14

Now, calculate the original mean.Meanoriginal=1064010Mean_{original} = \frac{10640}{10}Meanoriginal​=1010640​

STEP 15

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

STEP 16

STEP 17

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

STEP 18

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

STEP 19

Now, calculate the new mean.Meannew=Sum of all new rentsNumber of rentsMean_{new} = \frac{Sum\, of\, all\, new\, rents}{Number\, of\, rents}Meannew​=NumberofrentsSumofallnewrents​

STEP 20

Calculate the new mean.Meannew=1024010Mean_{new} = \frac{10240}{10}Meannew​=1010240​

STEP 21

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

STEP 22

Compare the original mean and the new mean to determine the change.Change in mean=Meannew−MeanoriginalChange\, in\, mean = Mean_{new} - Mean_{original}Changeinmean=Meannew​−Meanoriginal​

STEP 23

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

STEP 24

Calculate the change in mean.Change in mean=−40Change\, in\, mean = -40Changeinmean=−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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Monthly rents of 10 people change when one person's rent decreases from $1630 to$ 1230. How does the median and mean change?

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.
$Median_{original} = \frac{5th\, value + 6th\, value}{2}$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.