Answer :
Let's analyze the given data on the distribution of the number of motor vehicles per household to find the cumulative distribution, mean, median, and mode.
### Cumulative Distribution
To find the cumulative distribution in percentage, follow these steps:
1. Start with the proportion for 0 vehicles.
2. Add the next proportion for 1 vehicle to the previous cumulative sum.
3. Continue adding the proportions until all values are included.
4. Multiply each cumulative sum by 100 to convert them to percentages.
The calculations are as follows:
1. 0 vehicles: [tex]\( 0.19 \)[/tex]
2. 1 vehicle: [tex]\( 0.19 + 0.24 = 0.43 \)[/tex]
3. 2 vehicles: [tex]\( 0.43 + 0.32 = 0.75 \)[/tex]
4. 3 vehicles: [tex]\( 0.75 + 0.16 = 0.91 \)[/tex]
5. 4 vehicles: [tex]\( 0.91 + 0.08 = 0.99 \)[/tex]
6. 5 or more vehicles: [tex]\( 0.99 + 0.01 = 1.00 \)[/tex]
Next, we convert these cumulative sums into percentages by multiplying each by 100:
1. 0 vehicles: [tex]\( 0.19 \times 100 = 19\% \)[/tex]
2. 1 vehicle: [tex]\( 0.43 \times 100 = 43\% \)[/tex]
3. 2 vehicles: [tex]\( 0.75 \times 100 = 75\% \)[/tex]
4. 3 vehicles: [tex]\( 0.91 \times 100 = 91\% \)[/tex]
5. 4 vehicles: [tex]\( 0.99 \times 100 = 99\% \)[/tex]
6. 5 or more vehicles: [tex]\( 1.00 \times 100 = 100\% \)[/tex]
So, the cumulative distribution in percentages is:
[tex]\[ \begin{tabular}{ccc} \begin{tabular}{c} No. of \\ Motor Vehicles \end{tabular} & Proportion & Cumulative Percent \\ \hline 0 & 0.19 & 19\% \\ 1 & 0.24 & 43\% \\ 2 & 0.32 & 75\% \\ 3 & 0.16 & 91\% \\ 4 & 0.08 & 99\% \\ 5 (5 or more) & 0.01 & 100\% \end{tabular} \][/tex]
### Mean
To find the mean number of motor vehicles per household, multiply each number of vehicles by its proportion, then sum these products:
[tex]\[ \text{Mean} = (0 \times 0.19) + (1 \times 0.24) + (2 \times 0.32) + (3 \times 0.16) + (4 \times 0.08) + (5 \times 0.01) = 0 + 0.24 + 0.64 + 0.48 + 0.32 + 0.05 \][/tex]
[tex]\[ \text{Mean} = 0 + 0.24 + 0.64 + 0.48 + 0.32 + 0.05 = 1.73 \][/tex]
Thus, the mean number of motor vehicles per household is [tex]\(1.73\)[/tex] motor vehicles (rounded to two decimal places).
### Median
The median is the value that divides the data into two equal parts. Find the point where the cumulative proportion reaches or exceeds 0.5 (50%):
From the cumulative distribution, we see that the cumulative proportion reaches 50% between 1 and 2 vehicles. The first cumulative proportion greater than or equal to 0.5 is at 2 vehicles (75% cumulative).
Thus, the median number of motor vehicles is [tex]\(2\)[/tex] motor vehicles.
### Mode
The mode is the number of vehicles that appears most frequently. This corresponds to the highest proportion:
Here, the highest proportion is [tex]\(0.32\)[/tex] for 2 vehicles.
Thus, the mode number of motor vehicles is [tex]\(2\)[/tex] motor vehicles.
### Summary
The cumulative distribution, mean, median, and mode for the table are as follows:
#### Cumulative Distribution
[tex]\[ \begin{tabular}{ccc} \begin{tabular}{c} No. of \\ Motor Vehicles \end{tabular} & Proportion & Cumulative Percent \\ \hline 0 & 0.19 & 19\% \\ 1 & 0.24 & 43\% \\ 2 & 0.32 & 75\% \\ 3 & 0.16 & 91\% \\ 4 & 0.08 & 99\% \\ 5 (5 or more) & 0.01 & 100\% \end{tabular} \][/tex]
#### Mean
Mean [tex]\(= 1.73\)[/tex] motor vehicles
#### Median
Median [tex]\(= 2\)[/tex] motor vehicles
#### Mode
Mode [tex]\(= 2\)[/tex] motor vehicles
### Cumulative Distribution
To find the cumulative distribution in percentage, follow these steps:
1. Start with the proportion for 0 vehicles.
2. Add the next proportion for 1 vehicle to the previous cumulative sum.
3. Continue adding the proportions until all values are included.
4. Multiply each cumulative sum by 100 to convert them to percentages.
The calculations are as follows:
1. 0 vehicles: [tex]\( 0.19 \)[/tex]
2. 1 vehicle: [tex]\( 0.19 + 0.24 = 0.43 \)[/tex]
3. 2 vehicles: [tex]\( 0.43 + 0.32 = 0.75 \)[/tex]
4. 3 vehicles: [tex]\( 0.75 + 0.16 = 0.91 \)[/tex]
5. 4 vehicles: [tex]\( 0.91 + 0.08 = 0.99 \)[/tex]
6. 5 or more vehicles: [tex]\( 0.99 + 0.01 = 1.00 \)[/tex]
Next, we convert these cumulative sums into percentages by multiplying each by 100:
1. 0 vehicles: [tex]\( 0.19 \times 100 = 19\% \)[/tex]
2. 1 vehicle: [tex]\( 0.43 \times 100 = 43\% \)[/tex]
3. 2 vehicles: [tex]\( 0.75 \times 100 = 75\% \)[/tex]
4. 3 vehicles: [tex]\( 0.91 \times 100 = 91\% \)[/tex]
5. 4 vehicles: [tex]\( 0.99 \times 100 = 99\% \)[/tex]
6. 5 or more vehicles: [tex]\( 1.00 \times 100 = 100\% \)[/tex]
So, the cumulative distribution in percentages is:
[tex]\[ \begin{tabular}{ccc} \begin{tabular}{c} No. of \\ Motor Vehicles \end{tabular} & Proportion & Cumulative Percent \\ \hline 0 & 0.19 & 19\% \\ 1 & 0.24 & 43\% \\ 2 & 0.32 & 75\% \\ 3 & 0.16 & 91\% \\ 4 & 0.08 & 99\% \\ 5 (5 or more) & 0.01 & 100\% \end{tabular} \][/tex]
### Mean
To find the mean number of motor vehicles per household, multiply each number of vehicles by its proportion, then sum these products:
[tex]\[ \text{Mean} = (0 \times 0.19) + (1 \times 0.24) + (2 \times 0.32) + (3 \times 0.16) + (4 \times 0.08) + (5 \times 0.01) = 0 + 0.24 + 0.64 + 0.48 + 0.32 + 0.05 \][/tex]
[tex]\[ \text{Mean} = 0 + 0.24 + 0.64 + 0.48 + 0.32 + 0.05 = 1.73 \][/tex]
Thus, the mean number of motor vehicles per household is [tex]\(1.73\)[/tex] motor vehicles (rounded to two decimal places).
### Median
The median is the value that divides the data into two equal parts. Find the point where the cumulative proportion reaches or exceeds 0.5 (50%):
From the cumulative distribution, we see that the cumulative proportion reaches 50% between 1 and 2 vehicles. The first cumulative proportion greater than or equal to 0.5 is at 2 vehicles (75% cumulative).
Thus, the median number of motor vehicles is [tex]\(2\)[/tex] motor vehicles.
### Mode
The mode is the number of vehicles that appears most frequently. This corresponds to the highest proportion:
Here, the highest proportion is [tex]\(0.32\)[/tex] for 2 vehicles.
Thus, the mode number of motor vehicles is [tex]\(2\)[/tex] motor vehicles.
### Summary
The cumulative distribution, mean, median, and mode for the table are as follows:
#### Cumulative Distribution
[tex]\[ \begin{tabular}{ccc} \begin{tabular}{c} No. of \\ Motor Vehicles \end{tabular} & Proportion & Cumulative Percent \\ \hline 0 & 0.19 & 19\% \\ 1 & 0.24 & 43\% \\ 2 & 0.32 & 75\% \\ 3 & 0.16 & 91\% \\ 4 & 0.08 & 99\% \\ 5 (5 or more) & 0.01 & 100\% \end{tabular} \][/tex]
#### Mean
Mean [tex]\(= 1.73\)[/tex] motor vehicles
#### Median
Median [tex]\(= 2\)[/tex] motor vehicles
#### Mode
Mode [tex]\(= 2\)[/tex] motor vehicles
