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Question
Math
Posted 3 months ago

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 \\
\hlinex=x= room temperature (degrees Celsius) & xˉ=20.9\bar{x}=20.9 & sx=3s_{x}=3 \\
y=y= etching rate (micrometers per minute) & yˉ=6.6\bar{y}=6.6 & sy=0.7s_{y}=0.7 \\
& & r=0.99r=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.
y^=+x
\hat{y}=\square+\square x
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Answer from Sia
Posted 3 months ago
Solution by Steps
step 1
Calculate the slope (b) of the least-squares regression line using the formula b=r(sysx)b = r \left( \frac{s_y}{s_x} \right)
step 2
Substitute the given values r=0.99r = 0.99, sy=0.7s_y = 0.7, and sx=3s_x = 3 into the formula to find bb
step 3
Compute the slope: b=0.99(0.73)b = 0.99 \left( \frac{0.7}{3} \right)
step 4
Calculate the y-intercept (a) of the line using the formula a=yˉbxˉa = \bar{y} - b\bar{x}
step 5
Substitute the mean values yˉ=6.6\bar{y} = 6.6 and xˉ=20.9\bar{x} = 20.9 and the calculated slope bb into the formula to find aa
step 6
Compute the y-intercept: a=6.6b20.9a = 6.6 - b \cdot 20.9
step 7
Round the values of aa and bb 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 y^=a+bx\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.

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