Answer :
Alright class, let's begin by addressing each part of the question step-by-step. We'll work with the given sample data of tax-return preparation costs, which are:
[tex]\[120, 230, 110, 115, 160, 130, 150, 105, 195, 155, 105, 360, 120, 120, 140, 100, 115, 180, 235, 255\][/tex]
### Part (a): Compute the Mean, Median, and Mode
#### Mean:
The mean is the average of all the data points. To find the mean, sum all the values and then divide by the number of values.
[tex]\[ \text{Mean} = \frac{\sum \text{values}}{\text{number of values}} \][/tex]
Sum of values:
[tex]\[ 120 + 230 + 110 + 115 + 160 + 130 + 150 + 105 + 195 + 155 + 105 + 360 + 120 + 120 + 140 + 100 + 115 + 180 + 235 + 255 = 3200 \][/tex]
Number of values:
[tex]\[ 20 \][/tex]
Thus, the mean is:
[tex]\[ \text{Mean} = \frac{3200}{20} = 160 \][/tex]
#### Median:
The median is the middle value when the data points are arranged in ascending order. If the number of data points is even, the median is the average of the two middle numbers.
First, arrange the data in ascending order:
[tex]\[ 100, 105, 105, 110, 115, 115, 120, 120, 120, 130, 140, 150, 155, 160, 180, 195, 230, 235, 255, 360 \][/tex]
Since we have 20 data points, the median will be the average of the 10th and 11th values:
10th value: [tex]\( 130 \)[/tex]
11th value: [tex]\( 140 \)[/tex]
[tex]\[ \text{Median} = \frac{130 + 140}{2} = 135 \][/tex]
#### Mode:
The mode is the most frequently occurring value in the dataset. From the ordered data:
[tex]\[ 100, 105, 105, 110, 115, 115, 120, 120, 120, 130, 140, 150, 155, 160, 180, 195, 230, 235, 255, 360 \][/tex]
The value [tex]\( 120 \)[/tex] appears the most frequently (3 times).
[tex]\[ \text{Mode} = 120 \][/tex]
### Part (b): Compute the First and Third Quartiles
Quartiles divide the data into four equal parts.
#### First Quartile (Q1)
The first quartile is the median of the first half of the data (up to the 10th value).
First half of the data:
[tex]\[ 100, 105, 105, 110, 115, 115, 120, 120, 120, 130 \][/tex]
If we split this half, the median (Q1) is the average of the 5th and 6th values (115 and 115):
[tex]\[ \text{Q1} = \frac{115 + 115}{2} = 115 \][/tex]
#### Third Quartile (Q3)
The third quartile is the median of the second half of the data (from the 11th value on).
Second half of the data:
[tex]\[ 140, 150, 155, 160, 180, 195, 230, 235, 255, 360 \][/tex]
If we split this half, the median (Q3) is the average of the 5th and 6th values (180 and 195):
[tex]\[ \text{Q3} = \frac{180 + 195}{2} = 187.5 \][/tex]
### Part (c): Compute and Interpret the 90th Percentile
The nth percentile is a value below which a certain percentage of observations fall. The 90th percentile indicates that 90% of the data falls below this value.
To determine the 90th percentile, calculate the position using:
[tex]\[ P_{90} = 0.90 \times (n + 1) = 0.90 \times 21 = 18.9 \][/tex]
This implies the 90th percentile falls between the 18th and 19th values.
Values in order:
[tex]\[ 100, 105, 105, 110, 115, 115, 120, 120, 120, 130, 140, 150, 155, 160, 180, 195, 230, 235, 255, 360 \][/tex]
18th value: [tex]\( 235 \)[/tex]
19th value: [tex]\( 255 \)[/tex]
Interpolate between the 18th and 19th values:
[tex]\[ \text{90th Percentile} = 235 + 0.9 \times (255 - 235) = 235 + 0.9 \times 20 = 235 + 18 = 253 \][/tex]
### Interpretation
The 90th percentile indicates that 90% of the tax preparation costs are below \[tex]$253. In conclusion: a. Mean = \$[/tex]160, Median = \[tex]$135, Mode = \$[/tex]120
b. First Quartile (Q1) = \[tex]$115, Third Quartile (Q3) = \$[/tex]187.5
c. The 90th percentile is \$253.
[tex]\[120, 230, 110, 115, 160, 130, 150, 105, 195, 155, 105, 360, 120, 120, 140, 100, 115, 180, 235, 255\][/tex]
### Part (a): Compute the Mean, Median, and Mode
#### Mean:
The mean is the average of all the data points. To find the mean, sum all the values and then divide by the number of values.
[tex]\[ \text{Mean} = \frac{\sum \text{values}}{\text{number of values}} \][/tex]
Sum of values:
[tex]\[ 120 + 230 + 110 + 115 + 160 + 130 + 150 + 105 + 195 + 155 + 105 + 360 + 120 + 120 + 140 + 100 + 115 + 180 + 235 + 255 = 3200 \][/tex]
Number of values:
[tex]\[ 20 \][/tex]
Thus, the mean is:
[tex]\[ \text{Mean} = \frac{3200}{20} = 160 \][/tex]
#### Median:
The median is the middle value when the data points are arranged in ascending order. If the number of data points is even, the median is the average of the two middle numbers.
First, arrange the data in ascending order:
[tex]\[ 100, 105, 105, 110, 115, 115, 120, 120, 120, 130, 140, 150, 155, 160, 180, 195, 230, 235, 255, 360 \][/tex]
Since we have 20 data points, the median will be the average of the 10th and 11th values:
10th value: [tex]\( 130 \)[/tex]
11th value: [tex]\( 140 \)[/tex]
[tex]\[ \text{Median} = \frac{130 + 140}{2} = 135 \][/tex]
#### Mode:
The mode is the most frequently occurring value in the dataset. From the ordered data:
[tex]\[ 100, 105, 105, 110, 115, 115, 120, 120, 120, 130, 140, 150, 155, 160, 180, 195, 230, 235, 255, 360 \][/tex]
The value [tex]\( 120 \)[/tex] appears the most frequently (3 times).
[tex]\[ \text{Mode} = 120 \][/tex]
### Part (b): Compute the First and Third Quartiles
Quartiles divide the data into four equal parts.
#### First Quartile (Q1)
The first quartile is the median of the first half of the data (up to the 10th value).
First half of the data:
[tex]\[ 100, 105, 105, 110, 115, 115, 120, 120, 120, 130 \][/tex]
If we split this half, the median (Q1) is the average of the 5th and 6th values (115 and 115):
[tex]\[ \text{Q1} = \frac{115 + 115}{2} = 115 \][/tex]
#### Third Quartile (Q3)
The third quartile is the median of the second half of the data (from the 11th value on).
Second half of the data:
[tex]\[ 140, 150, 155, 160, 180, 195, 230, 235, 255, 360 \][/tex]
If we split this half, the median (Q3) is the average of the 5th and 6th values (180 and 195):
[tex]\[ \text{Q3} = \frac{180 + 195}{2} = 187.5 \][/tex]
### Part (c): Compute and Interpret the 90th Percentile
The nth percentile is a value below which a certain percentage of observations fall. The 90th percentile indicates that 90% of the data falls below this value.
To determine the 90th percentile, calculate the position using:
[tex]\[ P_{90} = 0.90 \times (n + 1) = 0.90 \times 21 = 18.9 \][/tex]
This implies the 90th percentile falls between the 18th and 19th values.
Values in order:
[tex]\[ 100, 105, 105, 110, 115, 115, 120, 120, 120, 130, 140, 150, 155, 160, 180, 195, 230, 235, 255, 360 \][/tex]
18th value: [tex]\( 235 \)[/tex]
19th value: [tex]\( 255 \)[/tex]
Interpolate between the 18th and 19th values:
[tex]\[ \text{90th Percentile} = 235 + 0.9 \times (255 - 235) = 235 + 0.9 \times 20 = 235 + 18 = 253 \][/tex]
### Interpretation
The 90th percentile indicates that 90% of the tax preparation costs are below \[tex]$253. In conclusion: a. Mean = \$[/tex]160, Median = \[tex]$135, Mode = \$[/tex]120
b. First Quartile (Q1) = \[tex]$115, Third Quartile (Q3) = \$[/tex]187.5
c. The 90th percentile is \$253.
