To determine which type of function best models the relationship between the day and the number of visitors, let's analyze the given dataset step-by-step:
The dataset contains:
- Days: [tex]\([1, 2, 3, 4, 5, 6, 7]\)[/tex]
- Visitors: [tex]\([45, 86, 124, 138, 145, 158, 162]\)[/tex]
We will consider linear and quadratic models and compare their effectiveness in modeling the data by examining the fit and corresponding errors.
### Step 1: Linear Model
A linear model has the form:
[tex]\[ V_{linear} = a \cdot day + b \][/tex]
Given the calculations, the linear model is determined as:
[tex]\[ a \approx 18.43, \quad b \approx 48.86 \][/tex]
Thus, the linear equation becomes:
[tex]\[ V_{linear} = 18.43 \cdot day + 48.86 \][/tex]
### Step 2: Quadratic Model
A quadratic model has the form:
[tex]\[ V_{quadratic} = a \cdot day^2 + b \cdot day + c \][/tex]
Given the calculations, the quadratic model is determined as:
[tex]\[ a \approx -3.86, \quad b \approx 49.29, \quad c \approx 2.57 \][/tex]
Thus, the quadratic equation becomes:
[tex]\[ V_{quadratic} = -3.86 \cdot day^2 + 49.29 \cdot day + 2.57 \][/tex]
### Step 3: Comparing the Sum of Squared Errors (SSE)
The Sum of Squared Errors (SSE) is a measure of how well a model's predictions match the observed data. It is calculated as:
[tex]\[ \text{SSE} = \sum (observed - predicted)^2 \][/tex]
Given the results:
- SSE for the linear model: [tex]\( \text{SSE}_{linear} \approx 1398.57 \)[/tex]
- SSE for the quadratic model: [tex]\( \text{SSE}_{quadratic} \approx 148.86 \)[/tex]
### Step 4: Selecting the Best Model
- A lower SSE indicates a better fit.
- The quadratic model's SSE (148.86) is significantly lower than the linear model's SSE (1398.57), indicating a much better fit.
- Additionally, we observe that the quadratic model has [tex]\( a \approx -3.86 \)[/tex], which is a negative value for [tex]\( a \)[/tex].
### Conclusion
Considering all the above points, the quadratic model with a negative value of [tex]\( a \)[/tex] provides the best fit for the given data.
Thus, the correct answer is:
C. a quadratic function with a negative value of a
