Use linear regression to find the equation for the linear function that best fits this data. Round both numbers to two decimal places. Write your final answer in a form of an equation
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To find the equation of the line that best fits the given data using linear regression, we need to compute the slope $m$ and the y-intercept $b$ of the line. The equation of the line is:

$y = mx + b$

Here, we are given the following data points:

$x: 1, 2, 3, 4, 5, 6$ $y: 70, 82, 102, 112, 141, 152$

Step 1: Calculate the Slope $m$

The formula for the slope $m$ in linear regression is:

$m = \frac{n\sum{x_i y_i} - \sum{x_i} \sum{y_i}}{n \sum{x_i^2} - (\sum{x_i})^2}$

where:

• $n$ is the number of data points,
• $x_i$ and $y_i$ are the values for $x$ and $y$,
• $\sum{x_i}$ is the sum of all $x$-values,
• $\sum{y_i}$ is the sum of all $y$-values,
• $\sum{x_i y_i}$ is the sum of the product of each pair of $x$ and $y$,
• $\sum{x_i^2}$ is the sum of the squares of all $x$-values.

Step 2: Calculate the Y-Intercept $b$

Once we have the slope $m$, the y-intercept $b$ is calculated by the formula:

$b = \frac{\sum{y_i} - m \sum{x_i}}{n}$

Let me calculate the values now.

Step 3: Perform the Calculation

We will first compute all the necessary sums:

• $\sum{x_i} = 1 + 2 + 3 + 4 + 5 + 6 = 21$
• $\sum{y_i} = 70 + 82 + 102 + 112 + 141 + 152 = 659$
• $\sum{x_i^2} = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 1 + 4 + 9 + 16 + 25 + 36 = 91$
• $\sum{x_i y_i} = (1 \times 70) + (2 \times 82) + (3 \times 102) + (4 \times 112) + (5 \times 141) + (6 \times 152) = 70 + 164 + 306 + 448 + 705 + 912 = 2605$

Now, applying these sums to the formulas for $m$ and $b$:

$m = \frac{6(2605) - (21)(659)}{6(91) - (21)^2}$ $m = \frac{15630 - 13839}{546 - 441}$ $m = \frac{1791}{105} = 17.06$

Now, calculate $b$:

$b = \frac{659 - 17.06(21)}{6}$ $b = \frac{659 - 358.26}{6} = \frac{300.74}{6} = 50.12$

Final Equation

The equation of the line that best fits the data is:

$y = 17.06x + 50.12$

Would you like more details on how these calculations work, or any other related concepts?

Here are some related questions you might find helpful:

1. How do you calculate the correlation coefficient to measure the strength of the linear relationship?
2. What is the significance of the slope and intercept in the context of this problem?
3. How can you graph the line of best fit using these values?
4. What is the difference between simple linear regression and multiple linear regression?
5. How does the least squares method work for finding the line of best fit?

Tip: To interpret the slope, it tells you how much the $y$-value changes for every 1-unit increase in $x$.