Chapter 12, Problem 1CRE

(a)

To determine

Explain: what the scatter plot says of the links between the cylindrical wall thickness and the gate speed.

Answer to Problem 1CRE

The relationship is positive, linear, moderately strong.

Explanation of Solution

Given:

  

  

Direction: Positive, as the plot slopes upward.

Form: Linear, since the points lie uniformly along a line and no curvature occurs.

Strength: Fairly solid, since the points are not too distant, nor not close together.

(b)

To determine

To find: the least-square regression line equation and specifying the variables that used.

Answer to Problem 1CRE

  $\widehat{y}=70.44+274.78x$

x the thickness, y the velocity

Explanation of Solution

Given:

  

  

The coefficients a and b are

  $\begin{array}{l}a=70.44\\ b=274.78\end{array}$

Calculation:

General regression equation of least-squares:

  $\widehat{y}=a+bx$

Regression equation of the least-squares

  $\widehat{y}=70.44+274.78x$

x the thickness, y the velocity

(c)

To determine

To explain: the regression line in part (b) overestimates or underestimates the gate speed of this cylinder.

Answer to Problem 1CRE

Overpredicted

Explanation of Solution

Given:

  

  

  $\begin{array}{l}x=0.4\\ y=104.8\end{array}$

From the part (b)

  $\widehat{y}=70.44+274.78x$

Calculation:

  $\widehat{y}=70.44+274.78x$

Putting the value of x with 0.4

  $\widehat{y}=70.44+274.78\left(0.4\right)=180.352$

It is observed then that the expected value of 180.352 is larger than 104.8 and therefore it is overpredicted the gate velocity.

(d)

To determine

Explain: the correct linear model in this case, explain the response with sufficient proof.

Answer to Problem 1CRE

Yes

Explanation of Solution

Given:

  

  

Yes, since there is no apparent trend on the residual plot and the residual in the plot is centred at about 0 on the basis of given plot.

(e)

To determine

To Explain: the given terms such that the slope, s, $r^2$ and the standard error of the slope.

Explanation of Solution

Given:

  

  

  $\begin{array}{l}b=274.78\\ s=56.3641\\ r^2=49.3\%\\ S{E}_b=88.18\end{array}$

(i) It is calculated that the slope will be 274.78, which means a gate speed rise of 274.78 per inch (of thickness) is required.

(ii) s= 56.3641 In other words, on average, 56,3641 assumptions are expected to vary from the true value.

(iii) $r^2$ = 49.3%. This means that the least-square regression line clarified 49.3 percent of the variation between the variables.

(iv)   $S{E}_b=88.18$ This means that the projected slope of the population regression line would on average differ 88.18.