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 \\
\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.
$\hat{y}=\square+\square x$
```

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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 $b = r \left( \frac{s_y}{s_x} \right)$

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.

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