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 linear function using linear regression, we can follow these steps:

Step 1: Calculate the Slope ($m$)

The formula for the slope is:

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

Where:

• $n$ is the number of data points,
• $\sum xy$ is the sum of the product of corresponding $x$ and $y$ values,
• $\sum x$ is the sum of the $x$-values,
• $\sum y$ is the sum of the $y$-values,
• $\sum x^2$ is the sum of the squares of the $x$-values.

Step 2: Calculate the Intercept ($b$)

The formula for the intercept is:

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

Now let's calculate each component:

Given dataTo find the equation of the linear function using linear regression, we can follow these steps:

Step 1: Calculate the Slope ($m$)

The formula for the slope is:

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

Where:

• $n$ is the number of data points,
• $\sum xy$ is the sum of the product of corresponding $x$ and $y$ values,
• $\sum x$ is the sum of the $x$-values,
• $\sum y$ is the sum of the $y$-values,
• $\sum x^2$ is the sum of the squares of the $x$-values.

Step 2: Calculate the Intercept ($b$)

The formula for the intercept is:

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

Now let's calculate each component:

Given

• $x$: 1, 2, 3, 4, 5, 6
• $y$: 78, 93, 114, 128, 153, 169

1. $\sum x = 1 + 2 + 3 + 4 + 5 + 6 = 21$
2. $\sum y = 78 + 93 + 114 + 128 + 153 + 169 = 735$
3. $\sum xy = (1 \times 78) + (2 \times 93) + (3 \times 114) + (4 \times 128) + (5 \times 153) + (6 \times 169) = 3369$
4. $\sum x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 91$

With these, we can now compute $m$ and $b$.

Step 3: Solve for $m$

$m = \frac{6(3369) - (21)(735)}{6(91) - (21)^2} = \frac{20214 - 15435}{546 - 441} = \frac{4779}{105} = 45.51$

Step 4: Solve for $b$

$b = \frac{735 - 45.51(21)}{6} = \frac{735 - 955.71}{6} = \frac{-220.71}{6} = -36.79$

Final Equation

So, the equation of the best fit line is:

$y = 45.51x - 36.79$

Would you like any more details or have questions?

Related Questions:

1. How would we calculate residuals from this regression?
2. What is the correlation coefficient for this data set?
3. How would the slope change if the data points were scaled?
4. How can this equation be used to predict future values?
5. What happens if one outlier is added to this data?

Tip: When performing linear regression, visualizing the data with the line can help identify potential outliers or nonlinear trends.