Question

Alexander uses cupric chloride to etch circuit boards. He recorded the room temperature, in ^\circ\text{C} ∘ Cdegrees, start text, C, end text, and the etching rate, in \dfrac{\mu\text{m}}{\text{min}} min μm ​ start fraction, mu, start text, m, end text, divided by, start text, m, i, n, end text, end fraction, 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 +\dfrac{1}{5} x y ^ ​ =2+ 5 1 ​ xy, with, hat, on top, equals, 2, plus, start fraction, 1, divided by, 5, end fraction, x What is the residual if the room temperature was 25^\circ\text{C}25 ∘ C25, degrees, start text, C, end text and the cupric chloride had an etching rate of 5\dfrac{\mu\text{m}}{\text{min}}5 min μm ​ 5, start fraction, mu, start text, m, end text, divided by, start text, m, i, n, end text, end fraction ?

Answer 1

The residual is the difference between the observed etching rate and the predicted etching rate

The residual of a room temperature of 25 degrees Celsius is $- 2\dfrac{\mu\text{m}}{\text{min}}$

How to determine the residual

The least squares regression equation for predicting the etching rate from the room temperature is given as:

$\hat y = 2 +\dfrac{1}{5} x$

When the room temperature is 25 degrees Celsius, the regression equation becomes

$\hat y = 2 +\dfrac{1}{5} *25$

Evaluate the product

$\hat y = 2 +5$

Evaluate the sum

$\hat y = 7$ --- this represents the predicted etching rate

The residual is calculated as:

Residual = Observed - Predicted

From the question, the observed etching rate is $5\dfrac{\mu\text{m}}{\text{min}}$

So, the residual equation becomes

$r = 5\dfrac{\mu\text{m}}{\text{min}} - 7\dfrac{\mu\text{m}}{\text{min}}$

Evaluate the difference

$r = - 2\dfrac{\mu\text{m}}{\text{min}}$

Hence, the residual of a room temperature of 25 degrees Celsius is $- 2\dfrac{\mu\text{m}}{\text{min}}$

Read more about residuals at:

https://brainly.com/question/2732316