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In the casting of metal parts, molten metal flows through a 鈥済ate鈥� 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 to automate the process.

Construct and interpret a $95\%$confidence interval for the slope of the population regression line. Explain how this interval is consistent with the results of Exercise R12.2.

Step by step solution

01

Given information

The given data is

02

Explanation

The given values are

$n=12$

$b=274.78$

$S{E}_b=88.18$

The degrees of freedom is the sample size decreased by 2 :

localid="1650647172335" $$\begin{gathered} df=n-2 \\ =12-2 \\ =10 \end{gathered}$$

The critical t-value can be found in table B in the row of $df=10$and in the column of $c=95\%$:

$t^{*}=2.228$

The boundaries of the confidence interval then become:

localid="1650647176471" $$\begin{gathered} b-t^{*}脳S{E}_b=274.78-2.228脳88.18 \\ =78.31496 \end{gathered}$$

localid="1650647181873" $$\begin{gathered} b+t^{*}脳S{E}_b=274.78+2.228脳88.18 \\ =471.24504 \end{gathered}$$

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