To find the equation for the linear function $y = mx + b$ that best fits the data, we use linear regression to compute the slope ($m$) and y-intercept ($b$). The formulas are:

$m = \frac{n \sum(xy) - \sum x \sum y}{n \sum(x^2) - (\sum x)^2}$

$b = \frac{\sum y - m \sum x}{n}$

Given Data:

$x = [1, 2, 3, 4, 5, 6]$ $y = [1194, 1631, 2593, 4110, 6089, 8569]$

Steps:

1. Compute the required sums: $\sum x, \sum y, \sum(xy), \sum(x^2)$.
2. Plug the values into the formulas for $m$ and $b$.

Let me calculate these values for you. numpy as np

Given data

x = np.array([1, 2, 3, 4, 5, 6]) y = np.array([1194, 1631, 2593, 4110, 6089, 8569])

Number of data points

n = len(x)

Calculate sums

sum_x = np.sum(x) sum_y = np.sum(y) sum_xy = np.sum(x * y) sum_x2 = np.sum(x**2)

Calculate slope (m) and intercept (b)

m = (n * sum_xy - sum_x * sum_y) / (n * sum_x2 - sum_x**2) b = (sum_y - m * sum_x) / n

m_rounded = round(m, 2) b_rounded = round(b, 2)

m_rounded, b_roundedThe equation of the linear function that best fits the given data is:

$y = 1479.03x - 1145.60$

This equation can be used to predict $y$ values for given $x$ values within this dataset's range.

Would you like a detailed breakdown of the calculations or any additional information?

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Related Questions:

1. How is the linear regression formula derived mathematically?
2. What is the correlation coefficient for this dataset?
3. How can we measure the goodness of fit for this linear model?
4. Could we use this equation for extrapolation beyond $x = 6$? Why or why not?
5. What are some practical applications of linear regression in data analysis?

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Tip:

Always check the residuals (differences between observed and predicted values) to evaluate how well a linear model fits the data.