Sure, let's go through each part step-by-step:
### Part A: Determining the Equation for the Line of Fit
To determine the equation for the line of fit (also known as the least squares regression line), we need to find the slope ([tex]\(m\)[/tex]) and intercept ([tex]\(b\)[/tex]) of the line that best fits the given data points. The general form of the equation of a line is:
[tex]\[ \text{Money Spent} = m \cdot \text{Hours} + b \][/tex]
Based on the calculations, we have:
- Slope ([tex]\(m\)[/tex]): 54.43756825627957
- Intercept ([tex]\(b\)[/tex]): -19.543137968693184
Therefore, the equation of the line of fit is:
[tex]\[ \text{Money Spent} = 54.43756825627957 \cdot \text{Hours} - 19.543137968693184 \][/tex]
### Part B: Identifying and Interpreting the Slope in the Context of This Scenario
The slope of the line is [tex]\(54.43756825627957\)[/tex].
Interpretation: The slope represents the rate of change of the money spent with respect to the hours spent at the hair salon. Specifically, it means that for every additional hour spent at the hair salon, the money spent increases by approximately [tex]$54.44.
### Part C: Predicting the Cost of a 6-Hour Appointment Using the Equation
Now, we will use the equation of the line of fit to predict the cost for a 6-hour hair salon appointment.
Step-by-Step:
1. Write down the equation of the line:
\[ \text{Money Spent} = 54.43756825627957 \cdot \text{Hours} - 19.543137968693184 \]
2. Substitute \( \text{Hours} = 6 \):
\[ \text{Money Spent} = 54.43756825627957 \cdot 6 - 19.543137968693184 \]
3. Calculate the result:
\[ \text{Money Spent} = 326.6254095376774 - 19.543137968693184 = 307.0822715689842 \]
Therefore, the predicted cost for a 6-hour hair salon appointment is approximately $[/tex]307.08.
