Sure, let's analyze the given data step by step.
### Step 1: Determine if there is a correlation
We first need to calculate the correlation coefficient between the x and y values.
The correlation coefficient for the given data is approximately [tex]\(0.7376\)[/tex]. This value indicates a moderate to strong positive correlation between the x and y values.
### Step 2: Identify the function family
Given that we want to find the best fit for the data, we analyze different types of functions. For this data set, a polynomial function fits well. In this case, we specifically found that a quadratic polynomial (second-degree polynomial) is appropriate.
### Step 3: Equation of the polynomial
The equation of the polynomial that describes the data can be written from the coefficients that were found. The equation is:
[tex]\[ y = 2967.65624054x^2 - 23570.00078929x + 34386.66256147 \][/tex]
### Step 4: Discussing the correlation coefficient
Since the function is polynomial and not linear, the correlation coefficient was used to initially check for any relationship. The moderate to strong positive correlation coefficient (~0.7376) indicates that as x increases, y increases in a somewhat proportional manner, though not perfectly due to the non-linearity.
### Step 5: Coefficient of determination
The coefficient of determination (denoted as [tex]\(R^2\)[/tex]) indicates how well the polynomial model fits the data. In this case, [tex]\(R^2\)[/tex] is approximately 0.8240.
### Step 6: Assessing the model's suitability for predictions
An [tex]\(R^2\)[/tex] value of 0.8240 suggests that about 82.4% of the variability in the y-values can be explained by the polynomial model. This is generally considered a good fit, meaning the model is reasonably accurate.
### Conclusion
- Correlation: Yes, there is a moderate to strong positive correlation.
- Function Family: Polynomial.
- Equation: [tex]\( y = 2967.65624054x^2 - 23570.00078929x + 34386.66256147 \)[/tex]
- Correlation Coefficient (if linear): The polynomial correlation coefficient wouldn't be applicable in the same sense, but if we still consider the linear correlation, it’s approximately 0.7376.
- What does this mean? The positive correlation indicates a relationship between x and y where increases in x are associated with increases in y.
- Coefficient of Determination: [tex]\(R^2 = 0.8240\)[/tex]
- Good for Predictions? Yes, because an [tex]\(R^2\)[/tex] value of 0.8240 indicates a strong explanatory power of the polynomial model, making it reliable for making predictions based on this data.
