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 aluminium 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. 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.

Do these data provide convincing evidence of a straight-line relationship between thickness and gate velocity in the population of pistons formed from this alloy of metal? Carry out an appropriate significance test at the $\alpha =0.05$level.

Step by step solution

01

Given information

The given data is

02

Explanation

The given data is

$n=12$

$b=274.78$

localid="1650534154906" $S{E}_b=88.18$

Determine the hypothesis

$H_0:\beta =0$

$H_a:\beta <0$

Compute the value of the test statistic

localid="1650647270417" $$\begin{gathered} t=\frac{b-{\beta }_0}{S{E}_b} \\ =\frac{274.78-0}{88.18} \\ \approx 3.116 \end{gathered}$$

The $P-$value is the chance of getting the test statistic's result, or a number that is more severe. The $p-$value is the number (or interval) in Table $B\text{'}$s column title that contains the row's t-value.

localid="1650647295982" $$\begin{gathered} df=n-2 \\ =12-2 \\ =10 \end{gathered}$$

$0.005<P<0.01$

The null hypothesis is rejected if the P-value is less than or equal to the significance level:

$P<0.05\Rightarrow Reject{H}_0$

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