Examine the data below for a stalk of corn.
| Day, $x$ | 9 | 12 | 22 | 40 |
|---|---|---|---|---|
| Height, $y$ (in) | 5 | 17 | 45 | 60 |
Use logarithmic regression to find an equation of the form $y=a+b \ln (x)$ to model the data.
$a=\square$
$b=\square$
Answer (2)
To model the corn stalk data, we calculated the coefficients for the logarithmic regression equation y = a + b ln ( x ) using the appropriate formulas. We found that a ≈ − 76.213 and b ≈ 37.673 , resulting in the equation y = − 76.213 + 37.673 ln ( x ) .
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Calculate the sums: ∑ ln ( x i ) , ∑ y i , ∑ ln ( x i ) y i , and ∑ ( ln ( x i ) ) 2 .
Use the formula b = n ( ∑ ( l n ( x i ) ) 2 ) − ( ∑ l n ( x i ) ) 2 n ( ∑ l n ( x i ) y i ) − ( ∑ l n ( x i )) ( ∑ y i ) to find b .
Use the formula a = n ( ∑ y i ) − b ( ∑ l n ( x i )) to find a .
The logarithmic regression equation is y = a + b ln ( x ) , where a = − 76.203 and b = 37.673 , so the equation is y = − 76.203 + 37.673 ln ( x ) .
a = − 76.203 , b = 37.673
Explanation
Understanding the Problem We are given a set of data points (x, y) representing the height of a corn stalk over time and are asked to find a logarithmic regression equation of the form y = a + b ln ( x ) that models this data. This means we need to find the values of the coefficients 'a' and 'b' that best fit the given data.
Formulas for a and b To find the values of 'a' and 'b', we will use the formulas for linear regression on the transformed data, where we replace x with ln ( x ) . The formulas are:
b = n ( ∑ ( l n ( x i ) ) 2 ) − ( ∑ l n ( x i ) ) 2 n ( ∑ l n ( x i ) y i ) − ( ∑ l n ( x i )) ( ∑ y i )
a = n ( ∑ y i ) − b ( ∑ l n ( x i ))
where n is the number of data points, which is 4 in this case.
Calculating Sums First, we calculate the necessary sums using the given data:
x i : [9, 12, 22, 40] y i : [5, 17, 45, 60] ln ( x i ) : [ ln ( 9 ) , ln ( 12 ) , ln ( 22 ) , ln ( 40 ) ] ≈ [2.197, 2.485, 3.091, 3.689]
∑ ln ( x i ) ≈ 2.197 + 2.485 + 3.091 + 3.689 = 11.462 ∑ y i = 5 + 17 + 45 + 60 = 127 ∑ ln ( x i ) y i ≈ ( 2.197 ) ( 5 ) + ( 2.485 ) ( 17 ) + ( 3.091 ) ( 45 ) + ( 3.689 ) ( 60 ) = 10.985 + 42.245 + 139.095 + 221.34 = 413.665 ∑ ( ln ( x i ) ) 2 ≈ ( 2.197 ) 2 + ( 2.485 ) 2 + ( 3.091 ) 2 + ( 3.689 ) 2 = 4.827 + 6.175 + 9.554 + 13.609 = 34.165
Calculating a and b Now, we substitute these sums into the formulas for 'a' and 'b':
b = 4 ( 34.165 ) − ( 11.462 ) 2 4 ( 413.665 ) − ( 11.462 ) ( 127 ) = 136.66 − 131.377 1654.66 − 1455.674 = 5.283 198.986 ≈ 37.673
a = 4 127 − 37.673 ( 11.462 ) = 4 127 − 431.85 = 4 − 304.85 ≈ − 76.213
Final Equation Therefore, the logarithmic regression equation is approximately:
y = − 76.203 + 37.673 ln ( x )
Thus, a ≈ − 76.203 and b ≈ 37.673 .
Examples
Logarithmic regression is useful in modeling phenomena where the rate of change decreases over time. For example, it can be used to model the growth of a tree, where the height increases rapidly at first but then slows down as the tree matures. Similarly, it can be used to model the decay of a radioactive substance, where the amount of substance decreases rapidly at first but then slows down as it approaches zero. In finance, logarithmic regression can model the relationship between stock prices and time, capturing the diminishing returns often observed in mature markets. Understanding these models helps in making predictions and informed decisions in various real-world scenarios.
