Question

In the casting of metal parts, molten metal flows through a “gate” into a die that shapes the part. The gate velocity (the speed at which metal is forced through the gate) plays a critical role in die casting. A firm that casts cylindrical aluminum pistons examined a random sample of $12$pistons formed from the same alloy of metal. What is the relationship between the cylinder wall thickness (inches) and the gate velocity (feet per second) chosen by the skilled workers who do the casting? If there is a clear pattern, it can be used to direct new workers or automate the process. A scatterplot of the data is shown below.

A least-squares regression analysis was performed on the data. Some computer output and a residual plot are shown below. A Normal probability plot of the residuals (not shown) is roughly linear.

$\begin{array}{lcccc}\text{Predictor}& \text{Coef}& \text{SE Coef}& T& \text{P}\\ \text{Constant}& 70.44& 52.90& 1.33& 0.213\\ \text{Thickness}& 274.78& 88.18& \star \star \star & \star \star \star \end{array}$

$S=56.3641\mathrm{R}-\mathrm{Sq}=49.3\%\mathrm{R}-\mathrm{Sq}\left(\mathrm{adj}\right)=44.2\%$

(a) Describe what the scatterplot tells you about the relationship between cylinder wall thickness and gate velocity.

(b) What is the equation of the least-squares regression line? Define any variables you use.

(c) One of the cylinders in the sample had a wall thickness of $0.4$inches. The gate velocity chosen for this cylinder was $104.8$feet per second. Does the regression line in part (b) overpredict or underpredict the gate velocity for this cylinder? By how much? Show your work.

(d) Is a linear model appropriate in this setting? Justify your answer with appropriate evidence.

(e) Interpret each of the following in context:

(i) The slope

(ii)$s$

(iii) $r^2$

(iv) The standard error of the slope

Short answer

(a) The scatter plot diagram is Positive, Linear, and fairly Solid.

(b) The regression line equation is $\hat{y}=70.44+274.78x$and the variables are $x$the thickness, $y$the velocity.

(c) The expected value of $180.352$is thus greater than $104.8$, indicating that the gate velocity has been overestimated.

(d) Yes, there is a linear model appropriate in this setting.

(e) The values of

(i) $b=274.78$

(ii)$s=56.3641$

(iii)$r^2=49.3\%$

(iv)$S{E}_b=88.18$

Step-by-step solution

Part(a) Step 1: Given Information

Part(a) Step 2: Explanation

The plot slopes upward, hence the direction is positive.

Because the points sit uniformly along a line and there is no curvature, the form is linear.

Strength: Good, because the points aren't too far apart or too near together.

Part(b) Step 1: Given Information

Part(b) Step 2: Explanation

The coefficient of $a$and $b$are:

$a=70.44$

$b=274.78$

General regression equation

$\hat{y}=a+bx$

Part(c) Step 1: Given Information

Part(c) Step 2: Explanation

$x=0.4$

$y=104.8$

From the part (b)

$\hat{y}=70.44+274.78x$

Replacing the value of $x$

$$\begin{gathered} \hat{y}=70.44+274.78(0.4) \\ =180.352 \end{gathered}$$

Part(d) Step 1: Given Information

Part(d) Step 2: Explanation

Yes, because the residual plot shows no discernible trend and the residual in the plot is centered at roughly $0$on the basis of the presented figure.

Part(e) Step 1: Given Information

Part(e) Step 2: Explanation

(i) The slope is estimated to be $274.78$, therefore a gate speed raise of $274.78$per inch of thickness is necessary.

(ii) $s=56.3641$In other words, $56,3641$assumptions are predicted to differ from the genuine value on average.

(iii) $r^2=49.3\%$This means the least-square regression line explained $49.3\%$of the variance between the variables.

(iv)$S{E}_b=88.18$This means that the population regression line's anticipated slope would deviate by an average of $88.18$.