Dmitri wants to look at the relationship between room temperature and the etching rate each time he has etched a circuit board with cupric chloride. The data show a linear pattern with the summary statistics shown below:
\begin{tabular}{lll}
& mean & standard deviation \
\hline$x=$ room temperature (degrees Celsius) & $\bar{x}=20.9$ & $s_{x}=3$ \
$y=$ etching rate (micrometers per minute) & $\bar{y}=6.6$ & $s_{y}=0.7$ \
& & $r=0.99$
\end{tabular}

Find the equation of the least-squares regression line for predicting the etching rate from the room temperature.
Round your entries to the nearest hundredth.
$$
\hat{y}=\square+\square x
$$

Question

Dmitri wants to look at the relationship between room temperature and the etching rate each time he has etched a circuit board with cupric chloride. The data show a linear pattern with the summary statistics shown below:
\begin{tabular}{lll} 
& mean & standard deviation \
\hline$x=$ room temperature (degrees Celsius) & $\bar{x}=20.9$ & $s_{x}=3$ \
$y=$ etching rate (micrometers per minute) & $\bar{y}=6.6$ & $s_{y}=0.7$ \
& & $r=0.99$
\end{tabular}

Find the equation of the least-squares regression line for predicting the etching rate from the room temperature.
Round your entries to the nearest hundredth.
$$
\hat{y}=\square+\square x
$$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Calculate the slope (b) of the least-squares regression line using the formula $b = r \left( \frac{s_y}{s_x} \right)$. ‖ ‖ step 2 ⋮ Substitute the given values $r = 0.99$, $s_y = 0.7$, and $s_x = 3$ into the formula to find $b$. ‖ ‖ step 3 ⋮ Compute the slope: $b = 0.99 \left( \frac{0.7}{3} \right)$. ‖ ‖ step 4 ⋮ Calculate the y-intercept (a) of the line using the formula $a = \bar{y} - b\bar{x}$. ‖ ‖ step 5 ⋮ Substitute the mean values $\bar{y} = 6.6$ and $\bar{x} = 20.9$ and the calculated slope $b$ into the formula to find $a$. ‖ ‖ step 6 ⋮ Compute the y-intercept: $a = 6.6 - b \cdot 20.9$. ‖ ‖ step 7 ⋮ Round the values of $a$ and $b$ to the nearest hundredth to get the final equation of the least-squares regression line. ‖ ∻Answer∻ ⚹ The equation of the least-squares regression line is $\hat{y} = a + bx$. ⚹ ∻Key Concept∻ ⚹Least-Squares Regression Line⚹ ∻Explanation∻ ⚹The least-squares regression line is used to predict the value of a dependent variable based on the value of an independent variable. The line is determined by its slope and y-intercept, which can be calculated using the mean values, standard deviations, and correlation coefficient of the data.⚹◊What is the formula for calculating the slope of the least-squares regression line in this scenario?⍭ Generate me a similar question◊

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