Question

Question: Defects in nuclear missile housing parts. The technique of multivariable testing (MVT) was discussed in the Journal of the Reliability Analysis Center (First Quarter, 2004). MVT was shown to improve the quality of carbon-foam rings used in nuclear missile housings. The rings are produced via a casting process that involves mixing ingredients, oven curing, and carving the finished part. One type of defect analyzed was the number y of black streaks in the manufactured ring. Two variables found to impact the number of defects were turntable speed (revolutions per minute),x₁, and cutting-blade position (inches from center), x₂.

1. The researchers discovered “an interaction between blade position and turntable speed.” Hypothesize a regression model for E(y) that incorporates this interaction.
2. The researchers reported a positive linear relationship between number of defects (y) and turntable speed (x₁)but found that the slope of the relationship was much steeper for lower values of cutting-blade position (x₂). What does this imply about the interaction term in the model, part a? Explain.

Short answer

Answer

1. A regression model for E(y) with x₁ and x₂ as functions while there is an interaction amongst the variable can be written as$E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_1{x}_2$
2. Since for lower value of cutting blade position the relationship between y and x₂ is much steeper and there is a positive relationship between y and this means that for lower values of x₁ and x₂ there is some relationship between x₁ and x₂ as the lines will interact at lower values of x₁ and x₂ .

Step-by-step solution

Definition

Multivariate testing is technique used for testing a hypothesis in which multiple variables are modified. This testing is used to determine which combination of variations performs well among all the possible combinations

Regression model for E(y)

b.

Since the researchers have observed an interaction amongst the blade position and the turntable speed, a new variable needs to be added in the model to represent this interaction in the model. A regression model for E(y) with x₁ and x₂ as functions while there is an interaction amongst the variable which can be written as $E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_1{x}_2$.

The hypothesis is that the rings are being produced via a casting process that involves mixing ingredients, oven curing, and carving the finished part, where one type of defect analyzed was the number y of black streaks in the manufactured ring and the two variables found to impact the number of defects were turntable speed (revolutions per minute),x₁ and cutting-blade position (inches from center), x₂.Therefore, the null and the alternative hypothesis are:$H_0\text{:}{\beta }_0={\beta }_1=0$

and At least one of the $\beta$should be zero respectively.

Interpretation of the interaction term

b.

Since for lower value of cutting blade position the relationship between y and x₂ is much steeper and there is a positive relationship between y and this means that for lower values of x₁ and x₂ there is some relationship between x₁ and x₂ as the lines will interact at lower values of x₁ and x₂ .