The table shows the relationship between the number of trucks filled with mulch $(x)$ and the number of tons of mulch $(y)$ delivered by a landscaping company. Which regression equation models the data?
A. $y=4.8 x+3$
B. $y=3 x+4.8$
C. $y=x+20.8$
D. $y=20.8 x+1$
\begin{tabular}{|c|c|c|c|}
\hline $x$ & $y$ & $x ^2$ & $x y$ \\
\hline 1.5 & 9.5 & 2.25 & 14.25 \\
\hline 3 & 17 & 9 & 51 \\
\hline 5.5 & 34.5 & 30.25 & 189.75 \\
\hline 6.25 & 31 & 39.06 & 193.75 \\
\hline 7 & 35 & 49 & 245 \\
\hline$\sum x \sim$ & \begin{tabular}{c}
$\sum y \sim$
127
\end{tabular} & \begin{tabular}{c}
$\sum x^2 \sim$
129.56
\end{tabular} & \begin{tabular}{c}
$\sum x y \sim$
693.75
\end{tabular} \\
\hline
\end{tabular}
Answer (2)
Calculate the slope b using the formula: b = n ( ∑ x 2 ) − ( ∑ x ) 2 n ( ∑ x y ) − ( ∑ x ) ( ∑ y ) ≈ 4.811 .
Calculate the y-intercept a using the formula: a = n ∑ y − b ( ∑ x ) ≈ 3.029 .
Form the regression equation: y = a + b x ≈ 3.029 + 4.811 x .
Choose the closest equation from the options: y = 4.8 x + 3 .
Explanation
Understanding the Problem We are given a table of x and y values representing the relationship between the number of trucks filled with mulch ( x ) and the number of tons of mulch delivered ( y ). Our goal is to find the regression equation that best models this data. The general form of a linear regression equation is y = a + b x , where b is the slope and a is the y-intercept. We will use the provided sums to calculate a and b .
Gathering the Data We have the following sums from the table:
∑ x = 23.25 ∑ y = 127 ∑ x 2 = 129.56 ∑ x y = 693.75
Also, we know that the number of data points is n = 5 .
Calculating the Slope The formula to calculate the slope b is:
b = n ( ∑ x 2 ) − ( ∑ x ) 2 n ( ∑ x y ) − ( ∑ x ) ( ∑ y )
Substituting the values, we get:
b = 5 ( 129.56 ) − ( 23.25 ) 2 5 ( 693.75 ) − ( 23.25 ) ( 127 )
b = 647.8 − 540.5625 3468.75 − 2952.75
b = 107.2375 516
b ≈ 4.811
Calculating the Y-Intercept The formula to calculate the y-intercept a is:
a = n ∑ y − b ( ∑ x )
Substituting the values, we get:
a = 5 127 − 4.811 ( 23.25 )
a = 5 127 − 111.85575
a = 5 15.14425
a ≈ 3.029
Finding the Regression Equation Now we can write the regression equation as:
y = 3.029 + 4.811 x
Comparing this equation with the given options, the closest one is y = 4.8 x + 3 .
Final Answer Therefore, the regression equation that models the data is approximately y = 4.8 x + 3 .
Examples
Regression equations are used in various real-world scenarios. For instance, a landscaping company can use a regression equation to predict the amount of mulch needed for a project based on the number of trucks filled. Similarly, in agriculture, farmers can predict crop yield based on factors like rainfall and fertilizer usage. In finance, analysts use regression to predict stock prices based on historical data and market trends. These models help in making informed decisions and optimizing resource allocation.
The regression equation that models the relationship between the number of trucks filled with mulch and the tons of mulch delivered is approximately y = 4.8 x + 3 . This was determined by calculating the slope and y-intercept from the given data. Therefore, the correct choice is option A: y = 4.8 x + 3 .
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