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 Regression" ], "new_concepts": [], "current_concepts": [ "Logarithmic Regression" ] } ## Set up the regression system Using the *Logarithmic Regression* knowledge point $$ \begin{aligned} &\text{Model: } y = a + b \ln(x)\ &\text{Data points } (x_i, y_i): (9, 5), (12, 17), (22, 45), (40, 60)\ &\text{Let } z_i = \ln(x_i):\ &z_1 = \ln(9) \approx 2.19722\ &z_2 = \ln(12) \approx 2.48491\ &z_3 = \ln(22) \approx 3.09104\ &z_4 = \ln(40) \approx 3.68888 \end{aligned} $$ ## Compute summary statistics Using the *Logarithmic Regression* knowledge point $$ \begin{aligned} &n = 4\ &\sum z_i \approx 11.46205\ &\bar{z} \approx 2.86551\ &\sum y_i = 5 + 17 + 45 + 60 = 127\ &\bar{y} = 31.75\ &\sum z_i^2 \approx 2.19722^2 + 2.48491^2 + 3.09104^2 + 3.68888^2 \approx 34.16641\ &\sum z_i y_i \approx 2.19722(5) + 2.48491(17) + 3.09104(45) + 3.68888(60) \approx 413.6591 \end{aligned} $$ ## Calculate regression coefficients Using the *Logarithmic Regression* knowledge point $$ \begin{aligned} &b = \frac{n \sum z_i y_i - (\sum z_i)(\sum y_i)}{n \sum z_i^2 - (\sum z_i)^2} \approx \frac{4(413.6591) - (11.46205)(127)}{4(34.16641) - (11.46205)^2} \approx \frac{1654.6364 - 1455.6804}{136.6656 - 131.3786} \approx \frac{198.9560}{5.2870} \approx 37.6312\ &a = \bar{y} - b \bar{z} \approx 31.75 - 37.6312(2.86551) \approx 31.75 - 107.8346 \approx -76.0846 \end{aligned} $$ ## Round to two decimal places Using the *Logarithmic Regression* knowledge point $$ \begin{aligned} &a \approx -76.08\ &b \approx 37.63 \end{aligned} $$ Use logarithmic regression to find an equation of the form $y = a + b \ln(x)$ to model the data. $a =$ -76.08 $b =$ 37.63 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Statistics", "Logarithmic Regression" ] }

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QUESTION IMAGE

Question

Explanation:

Set up the regression system

Compute summary statistics

Calculate regression coefficients

Round to two decimal places

Answer: