examine the data below for a stalk of corn.

\begin{tabular}{|l|c|c|c|c|}
\hline
day, $x$ & 9 & 12 & 22 & 40 \\
\hline
height, $y$ (in) & 5 & 17 & 45 & 60 \\
\hline
\end{tabular}

use logarithmic regression to find an equation of the form $y = a + b \ln(x)$ to model the data.

$a =$
$b =$

Question

examine the data below for a stalk of corn.

\begin{tabular}{|l|c|c|c|c|}
\hline
day, $x$ & 9 & 12 & 22 & 40 \\
\hline
height, $y$ (in) & 5 & 17 & 45 & 60 \\
\hline
\end{tabular}

use logarithmic regression to find an equation of the form $y = a + b \ln(x)$ to model the data.

$a =$
$b =$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Logarithmic Functions", "Least Squares Regression" ], "new_concepts": [ "Logarithmic Regression" ], "current_concepts": [ "Logarithmic Regression", "Least Squares Regression", "Logarithmic Functions" ] } ## Transform the independent variable Using the *Logarithmic Functions* knowledge point $$ \begin{aligned} &x_1 = 9 \implies \ln(x_1) \approx 2.19722\ &x_2 = 12 \implies \ln(x_2) \approx 2.48491\ &x_3 = 22 \implies \ln(x_3) \approx 3.09104\ &x_4 = 40 \implies \ln(x_4) \approx 3.68888 \end{aligned} $$ ## Calculate summary statistics Using the *Least Squares Regression* knowledge point $$ \begin{aligned} &\sum \ln(x) \approx 11.46205\ &\sum y = 5 + 17 + 45 + 60 = 127\ &\sum (\ln(x))^2 \approx 34.31604\ &\sum y^2 = 5^2 + 17^2 + 45^2 + 60^2 = 5939\ &\sum y \ln(x) \approx 5 + 17(2.48491) + 45(3.09104) + 60(3.68888) \approx 407.67305 \end{aligned} $$ ## Solve for regression coefficients Using the *Least Squares Regression* knowledge point $$ \begin{aligned} &n = 4\ &b = \frac{n \sum y \ln(x) - (\sum \ln(x))(\sum y)}{n \sum (\ln(x))^2 - (\sum \ln(x))^2} \approx \frac{4(407.67305) - (11.46205)(127)}{4(34.31604) - (11.46205)^2} \approx 37.83\ &a = \frac{\sum y - b \sum \ln(x)}{n} \approx \frac{127 - 37.832(11.46205)}{4} \approx -76.65 \end{aligned} $$ 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. $a =$ -76.65 $b =$ 37.83 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Logarithmic Regression" ] }

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QUESTION IMAGE

Question

Explanation:

Transform the independent variable

Calculate summary statistics

Solve for regression coefficients

Answer: