91Ó°ÊÓ

The variable type of orange juice extractor (brand A or brand B) is a qualitative variable and the variable size of orange is a quantitative variable.

A model equation to describe the relationship between E(y) and size of orange and types of orange juice extractor used can be written as, $E\left(y\right)={\mathrm{β}}_0+{\mathrm{β}}_1{x}_1+{\mathrm{β}}_2{x}_2$ where $x_1=sizeoforange$ and $x_2=typeoforangejuiceextractorused$.

A model equation to describe the relationship between E(y) and size of orange and types of orange juice extractor used the slopes are different where can be written as $E\left(y\right)={\mathrm{β}}_0+{\mathrm{β}}_1{x}_1+{\mathrm{β}}_2{x}_2+{\mathrm{β}}_3{x}_3$, where $x_1=sizeoforange$and $x_2=typeoforangejuiceextractorused$.

A model graph without interaction can be drawn as two straight parallel lines where one line represents the relation between y and x₁ and the other line represents the relation between y and x₂. Since there is no relation amongst the variables, the lines do not interact.

A model graph with interaction can be drawn as two straight lines with interaction where one line represents the relation between y and x₁ and the other line represents the relation between y and x₂.

To check whether the models described in part b and part c are able to explain the variation in the data, F-test or goodness of fit test is conducted

$H_0:{\mathrm{β}}_1={\mathrm{β}}_2={\mathrm{β}}_3=0$ while$H_a:$ At least one of the parameters ${\mathrm{β}}_1,{\mathrm{β}}_2$and ${\mathrm{β}}_3$is nonzero.

To do goodness of fit testis conducted to check the overall goodness of the model.

$F-statistic=\frac{R^2/k}{\left(1-R^2\right)/\left[n-\left(k+1\right)\right]}$

Where,$SSE=∑{\left(y_i-\hat{y}\right)}^2$

.$$\begin{gathered} N=no.ofobservations \\ K=no.ofvariablesusedinthe\mathrm{mod}el\left(including{\mathrm{β}}_0\right) \end{gathered}$$

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