y

To find the linear regression equation that best fits the provided data points, we can use the formula for a linear regression line, which is given by:

$y = mx + b$

where $m$ is the slope and $b$ is the y-intercept.

Given Data Points:

• $x: [1, 2, 3, 4, 5, 6]$
• $y: [553, 5449, 19372, 49543, 112085, 190802]$

Steps to Calculate:

1. Calculate the means of $x$ and $y$: $\bar{x} = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = 3.5$ $\bar{y} = \frac{553 + 5449 + 19372 + 49543 + 112085 + 190802}{6} = 61100.67$

2. Calculate the slope $m$: $m = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}$

3. Calculate the y-intercept $b$: $b = \bar{y} - m \bar{x}$

Calculation:

1. Numerator and Denominator for $m$:
   • Compute $(x_i - \bar{x})(y_i - \bar{y})$ and $(x_i - \bar{x})^2$:

$x$	$y$	$x - \bar{x}$	$y - \bar{y}$	$(x - \bar{x})(y - \bar{y})$	$(x - \bar{x})^2$
1	553	-2.5	-60547.67	151368.675	6.25
2	5449	-1.5	-55651.67	83477.505	2.25
3	19372	-0.5	-41728.67	20864.335	0.25
4	49543	0.5	-11557.67	-5778.335	0.25
5	112085	1.5	50984.33	76476.495	2.25
6	190802	2.5	129701.33	324252.325	6.25

2. Sum up the columns: $\sum{(x - \bar{x})(y - \bar{y})} = 151368.675 + 83477.505 + 20864.335 - 5778.335 + 76476.495 + 324252.325 = 494661.005$ $\sum{(x - \bar{x})^2} = 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 = 17.5$

3. Calculate $m$: $m = \frac{494661.005}{17.5} \approx 28293.53$

4. Calculate $b$: $b = 61100.67 - (28293.53 \times 3.5) \approx -1241.75$

Final Linear Regression Equation:

$y = 28293.53x - 1241.75$

Rounding to two decimal places: $y \approx 28293.53x - 1241.75$

Summary:

The equation for the linear function that best fits the data is: $y = 28293.53x - 1241.75$

Would you like further details on any step, or do you have any questions? Here are some related questions you might consider:

1. How can I interpret the slope and intercept of this linear regression equation?
2. What are the limitations of using linear regression for this data?
3. How would you check the goodness of fit for this regression model?
4. What other regression techniques could be applied to this dataset?
5. How does increasing the number of data points affect the regression line?

Tip: Always visualize your data and the regression line to better understand the fit and any potential outliers.