Answer :
To match the bill amount of $82.60 with the corresponding call minutes, let's follow a detailed, step-by-step approach:
1. Identify the Relationship: We start by establishing the relationship between the call minutes and the bill amount. Given the linear nature of the data, we assume a linear model of the form:
[tex]\[ \text{Bill Amount} = m \times \text{Call Minutes} + c \][/tex]
where [tex]\( m \)[/tex] is the rate per minute and [tex]\( c \)[/tex] is the fixed rental charge.
2. Calculate the Coefficients:
Using the data points provided:
- Call Minutes: [tex]\([125, 150, 175, 200]\)[/tex]
- Bill Amounts: [tex]\([75, 85, 95, 105]\)[/tex]
By fitting a linear model, the coefficients were found to be:
[tex]\[ m = 0.4, \quad c = 25.0 \][/tex]
This gives us the linear equation:
[tex]\[ \text{Bill Amount} = 0.4 \times \text{Call Minutes} + 25 \][/tex]
3. Establish the Target Bill Amount:
We need to find the call minutes corresponding to the bill amount of [tex]\(82.60\)[/tex]:
[tex]\[ \text{Target Amount} = 82.60 \][/tex]
4. Compute the Residuals:
To determine which call minutes fit best with the target bill amount, calculate the residuals for each set of call minutes. A residual is the absolute difference between the calculated bill amount and the target amount.
Using the equation [tex]\( \text{Bill Amount} = 0.4 \times \text{Call Minutes} + 25 \)[/tex]:
- For [tex]\( 125 \)[/tex] call minutes:
[tex]\[ \text{Bill Amount} = 0.4 \times 125 + 25 = 50 + 25 = 75 \][/tex]
Residual: [tex]\( |75 - 82.60| = 7.60 \)[/tex]
- For [tex]\( 150 \)[/tex] call minutes:
[tex]\[ \text{Bill Amount} = 0.4 \times 150 + 25 = 60 + 25 = 85 \][/tex]
Residual: [tex]\( |85 - 82.60| = 2.40 \)[/tex]
- For [tex]\( 175 \)[/tex] call minutes:
[tex]\[ \text{Bill Amount} = 0.4 \times 175 + 25 = 70 + 25 = 95 \][/tex]
Residual: [tex]\( |95 - 82.60| = 12.40 \)[/tex]
- For [tex]\( 200 \)[/tex] call minutes:
[tex]\[ \text{Bill Amount} = 0.4 \times 200 + 25 = 80 + 25 = 105 \][/tex]
Residual: [tex]\( |105 - 82.60| = 22.40 \)[/tex]
5. Find the Closest Match:
Among the residuals computed:
- [tex]\( 7.60 \)[/tex] for [tex]\( 125 \)[/tex] call minutes
- [tex]\( 2.40 \)[/tex] for [tex]\( 150 \)[/tex] call minutes
- [tex]\( 12.40 \)[/tex] for [tex]\( 175 \)[/tex] call minutes
- [tex]\( 22.40 \)[/tex] for [tex]\( 200 \)[/tex] call minutes
The smallest residual is [tex]\( 2.40 \)[/tex], which corresponds to [tex]\( 150 \)[/tex] call minutes. Therefore, the call minutes that correspond most closely to the bill amount of [tex]\(82.60\)[/tex] is:
[tex]\[ \boxed{150} \][/tex]
1. Identify the Relationship: We start by establishing the relationship between the call minutes and the bill amount. Given the linear nature of the data, we assume a linear model of the form:
[tex]\[ \text{Bill Amount} = m \times \text{Call Minutes} + c \][/tex]
where [tex]\( m \)[/tex] is the rate per minute and [tex]\( c \)[/tex] is the fixed rental charge.
2. Calculate the Coefficients:
Using the data points provided:
- Call Minutes: [tex]\([125, 150, 175, 200]\)[/tex]
- Bill Amounts: [tex]\([75, 85, 95, 105]\)[/tex]
By fitting a linear model, the coefficients were found to be:
[tex]\[ m = 0.4, \quad c = 25.0 \][/tex]
This gives us the linear equation:
[tex]\[ \text{Bill Amount} = 0.4 \times \text{Call Minutes} + 25 \][/tex]
3. Establish the Target Bill Amount:
We need to find the call minutes corresponding to the bill amount of [tex]\(82.60\)[/tex]:
[tex]\[ \text{Target Amount} = 82.60 \][/tex]
4. Compute the Residuals:
To determine which call minutes fit best with the target bill amount, calculate the residuals for each set of call minutes. A residual is the absolute difference between the calculated bill amount and the target amount.
Using the equation [tex]\( \text{Bill Amount} = 0.4 \times \text{Call Minutes} + 25 \)[/tex]:
- For [tex]\( 125 \)[/tex] call minutes:
[tex]\[ \text{Bill Amount} = 0.4 \times 125 + 25 = 50 + 25 = 75 \][/tex]
Residual: [tex]\( |75 - 82.60| = 7.60 \)[/tex]
- For [tex]\( 150 \)[/tex] call minutes:
[tex]\[ \text{Bill Amount} = 0.4 \times 150 + 25 = 60 + 25 = 85 \][/tex]
Residual: [tex]\( |85 - 82.60| = 2.40 \)[/tex]
- For [tex]\( 175 \)[/tex] call minutes:
[tex]\[ \text{Bill Amount} = 0.4 \times 175 + 25 = 70 + 25 = 95 \][/tex]
Residual: [tex]\( |95 - 82.60| = 12.40 \)[/tex]
- For [tex]\( 200 \)[/tex] call minutes:
[tex]\[ \text{Bill Amount} = 0.4 \times 200 + 25 = 80 + 25 = 105 \][/tex]
Residual: [tex]\( |105 - 82.60| = 22.40 \)[/tex]
5. Find the Closest Match:
Among the residuals computed:
- [tex]\( 7.60 \)[/tex] for [tex]\( 125 \)[/tex] call minutes
- [tex]\( 2.40 \)[/tex] for [tex]\( 150 \)[/tex] call minutes
- [tex]\( 12.40 \)[/tex] for [tex]\( 175 \)[/tex] call minutes
- [tex]\( 22.40 \)[/tex] for [tex]\( 200 \)[/tex] call minutes
The smallest residual is [tex]\( 2.40 \)[/tex], which corresponds to [tex]\( 150 \)[/tex] call minutes. Therefore, the call minutes that correspond most closely to the bill amount of [tex]\(82.60\)[/tex] is:
[tex]\[ \boxed{150} \][/tex]
