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Math

QuestionFind the slope of the best fit line for clubhead speeds (mph) and distances (yards) from 20 games data.

Studdy Solution

STEP 1

Assumptions1. The data provided is accurate and complete. . The relationship between clubhead speed and distance is linear.
3. The best fit line is found using the method of least squares, which minimizes the sum of the squares of the residuals (the differences between the observed and predicted values).

STEP 2

First, we need to calculate the mean (average) of the clubhead speeds and the mean of the distances. The formula for the mean of a set of numbers is the sum of the numbers divided by the count of the numbers.
Mean of Clubhead Speed=Sum of Clubhead SpeedsCount of Clubhead Speeds\text{Mean of Clubhead Speed} = \frac{\text{Sum of Clubhead Speeds}}{\text{Count of Clubhead Speeds}}Mean of Distance=Sum of DistancesCount of Distances\text{Mean of Distance} = \frac{\text{Sum of Distances}}{\text{Count of Distances}}

STEP 3

Next, we calculate the differences between each clubhead speed and the mean clubhead speed, and between each distance and the mean distance. We'll call these differences "deviations".
Deviation of Clubhead Speed=Clubhead Speed−Mean of Clubhead Speed\text{Deviation of Clubhead Speed} = \text{Clubhead Speed} - \text{Mean of Clubhead Speed}Deviation of Distance=Distance−Mean of Distance\text{Deviation of Distance} = \text{Distance} - \text{Mean of Distance}

STEP 4

Then, we calculate the product of the deviations for each pair of clubhead speed and distance.
Product of Deviations=Deviation of Clubhead Speed×Deviation of Distance\text{Product of Deviations} = \text{Deviation of Clubhead Speed} \times \text{Deviation of Distance}

STEP 5

We also need to calculate the square of the deviation of clubhead speed.
Square of Deviation of Clubhead Speed=(Deviation of Clubhead Speed)2\text{Square of Deviation of Clubhead Speed} = (\text{Deviation of Clubhead Speed})^2

STEP 6

The slope of the best fit line (also known as the regression coefficient) is the sum of the products of deviations divided by the sum of the squares of deviations of clubhead speed.
lope=Sum of Products of DeviationsSum of Squares of Deviations of Clubhead Speed\text{lope} = \frac{\text{Sum of Products of Deviations}}{\text{Sum of Squares of Deviations of Clubhead Speed}}

STEP 7

Now, we plug in the calculated values into the formula for the slope and calculate the slope.

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