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Answer :
To find the equation of the regression line for the given data, we use linear regression. Linear regression finds the best-fitting straight line through the points by minimizing the sum of the squared differences between the observed values and those predicted by the line. The equation of this line is usually expressed in the form:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the intercept.
For this data, the following steps outline the process to determine the regression line:
1. List the data points: We have pairs of data [tex]\((x, y)\)[/tex], where [tex]\( x \)[/tex] is the age and [tex]\( y \)[/tex] is the percent of maximum throwing speed.
2. Compute the slope ([tex]\( m \)[/tex]): This involves using sums of the data points. The formula for the slope [tex]\( m \)[/tex] is given as:
[tex]\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \][/tex]
where [tex]\( n \)[/tex] is the number of points, [tex]\( \sum xy \)[/tex] is the sum of products of corresponding [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values, [tex]\( \sum x \)[/tex] is the sum of [tex]\( x \)[/tex] values, [tex]\( \sum y \)[/tex] is the sum of [tex]\( y \)[/tex] values, and [tex]\( \sum x^2 \)[/tex] is the sum of squared [tex]\( x \)[/tex] values.
3. Compute the intercept ([tex]\( b \)[/tex]): The formula for the intercept [tex]\( b \)[/tex] is:
[tex]\[ b = \frac{(\sum y)(\sum x^2) - (\sum x)(\sum xy)}{n(\sum x^2) - (\sum x)^2} \][/tex]
4. Substitute the slope and intercept into the equation: Once you have calculated [tex]\( m \)[/tex] and [tex]\( b \)[/tex], substitute them into the linear equation [tex]\( y = mx + b \)[/tex].
Based on the calculations, the equation of the regression line for this data set is:
[tex]\[ y = 1.99x + 42.62 \][/tex]
This means for every additional year of age, the percent of maximum throwing speed increases by approximately 1.99%, starting at a base value of 42.62% when age is zero.
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the intercept.
For this data, the following steps outline the process to determine the regression line:
1. List the data points: We have pairs of data [tex]\((x, y)\)[/tex], where [tex]\( x \)[/tex] is the age and [tex]\( y \)[/tex] is the percent of maximum throwing speed.
2. Compute the slope ([tex]\( m \)[/tex]): This involves using sums of the data points. The formula for the slope [tex]\( m \)[/tex] is given as:
[tex]\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \][/tex]
where [tex]\( n \)[/tex] is the number of points, [tex]\( \sum xy \)[/tex] is the sum of products of corresponding [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values, [tex]\( \sum x \)[/tex] is the sum of [tex]\( x \)[/tex] values, [tex]\( \sum y \)[/tex] is the sum of [tex]\( y \)[/tex] values, and [tex]\( \sum x^2 \)[/tex] is the sum of squared [tex]\( x \)[/tex] values.
3. Compute the intercept ([tex]\( b \)[/tex]): The formula for the intercept [tex]\( b \)[/tex] is:
[tex]\[ b = \frac{(\sum y)(\sum x^2) - (\sum x)(\sum xy)}{n(\sum x^2) - (\sum x)^2} \][/tex]
4. Substitute the slope and intercept into the equation: Once you have calculated [tex]\( m \)[/tex] and [tex]\( b \)[/tex], substitute them into the linear equation [tex]\( y = mx + b \)[/tex].
Based on the calculations, the equation of the regression line for this data set is:
[tex]\[ y = 1.99x + 42.62 \][/tex]
This means for every additional year of age, the percent of maximum throwing speed increases by approximately 1.99%, starting at a base value of 42.62% when age is zero.
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