Table 4 shows the yield (in bushels per acre) and the total...

Table 4 shows the yield (in bushels per acre) and the total production (in millions of bushels) for corn in the United States for selected years since $1950 .$ Let $x$ $$ \begin{array}{lrcc} \hline & & \text { Yield (bushels } & \text { Total Production } \\ \text { Year } & x & \text { per acre) } & \text { (million bushels) } \\ 1950 & 50 & 38 & 2,782 \\ 1960 & 60 & 56 & 3,479 \\ 1970 & 70 & 81 & 4,802 \\ 1980 & 80 & 98 & 6,867 \\ 1990 & 90 & 116 & 7,802 \\ 2000 & 100 & 140 & 10,192 \\ 2010 & 110 & 153 & 12,447 \\ \hline \end{array} $$ represent years since $1900 .$ Find a logarithmic regression model $(y=a+b \ln x)$ for the yield. Estimate (to the nearest bushel per acre) the yield in 2024

Table 4 shows the yield (in bushels per acre) and the total production (in millions of bushels) for corn in the United States for selected years since $1950 .$ Let $x$
$$
\begin{array}{lrcc}
\hline & & \text { Yield (bushels } & \text { Total Production } \\
\text { Year } & x & \text { per acre) } & \text { (million bushels) } \\
1950 & 50 & 38 & 2,782 \\
1960 & 60 & 56 & 3,479 \\
1970 & 70 & 81 & 4,802 \\
1980 & 80 & 98 & 6,867 \\
1990 & 90 & 116 & 7,802 \\
2000 & 100 & 140 & 10,192 \\
2010 & 110 & 153 & 12,447 \\
\hline
\end{array}
$$
represent years since $1900 .$ Find a logarithmic regression model $(y=a+b \ln x)$ for the yield. Estimate (to the nearest bushel per acre) the yield in 2024

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Key Concepts

Logarithmic Regression

A regression model where the natural logarithm of the independent variable is used in the equation to capture nonlinear relationships between variables. This transformation often linearizes data that grows at a decreasing rate, making the model easier to interpret and fit using linear techniques.

Least Squares Estimation

A statistical method employed to determine the best-fit parameters (such as a and b in the logarithmic regression model) by minimizing the sum of the squares of the residuals, which are the differences between observed and predicted values.

Predictive Modeling

The procedure of using a fitted regression model to estimate future or unseen data points. By substituting a new value of the independent variable into the regression equation, one can forecast the corresponding value of the dependent variable.

Data Transformation

The process of applying a mathematical function, like the logarithm, to the independent variable to stabilize variance, improve the linearity of the relationship, and meet the assumptions required for linear regression analysis.

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