Question

alexander uses cupric chloride to etch circuit boards. he recorded the room temperature, in °c, and the etching rate, in \\(\frac{\mu\text{m}}{\text{min}}\\), of the cupric chloride. after plotting his results, alexander noticed that the relationship between the two variables was fairly linear, so he used the data to calculate the following least squares regression equation for predicting the etching rate from the room temperature: \\(\hat{y} = 2 + \frac{1}{5}x\\) what is the residual if the room temperature was \\(25^\circ\text{c}\\) and the cupric chloride had an etching rate of \\(5\frac{\mu\text{m}}{\text{min}}\\)?

Explanation:

Step1: Recall the formula for residual

The residual is calculated as the actual value minus the predicted value, i.e., \( \text{Residual} = y - \hat{y} \), where \( y \) is the actual etching rate and \( \hat{y} \) is the predicted etching rate from the regression equation.

Step2: Identify the values of \( x \), \( y \), and the regression equation

We know that the room temperature \( x = 25^\circ \text{C} \), the actual etching rate \( y = 5 \frac{\mu\text{m}}{\text{min}} \), and the regression equation is \( \hat{y} = 2 + \frac{1}{5}x \).

Step3: Calculate the predicted value \( \hat{y} \)

Substitute \( x = 25 \) into the regression equation:
\[
\hat{y} = 2 + \frac{1}{5} \times 25
\]
First, calculate \( \frac{1}{5} \times 25 = 5 \), then \( \hat{y} = 2 + 5 = 7 \).

Step4: Calculate the residual

Using the residual formula \( \text{Residual} = y - \hat{y} \), substitute \( y = 5 \) and \( \hat{y} = 7 \):
\[
\text{Residual} = 5 - 7 = -2
\]

Answer:

The residual is \(-2\) \( \frac{\mu\text{m}}{\text{min}} \).