Question

Question: Comparing two orange juice extractors. The Florida Citrus Commission is interested in evaluating the performance of two orange juice extractors, brand A and brand B. It is believed that the size of the fruit used in the test may influence the juice yield (amount of juice per pound of oranges) obtained by the extractors. The commission wants to develop a regression model relating the mean juice yield E(y) to the type of orange juice extractor (brand A or brand B) and the size of orange (diameter), .

a. Identify the independent variables as qualitative or quantitative.

b. Write a model that describes the relationship between E(y) and size of orange as two parallel lines, one for each brand of extractor.

c. Modify the model of part b to permit the slopes of the two lines to differ.

d. Sketch typical response lines for the model of part b. Do the same for the model of part c. Carefully label your graphs.

e. Specify the null and alternative hypotheses you would use to determine whether the model in part c provides more information for predicting yield than does the model in part b.

f. Explain how you would obtain the quantities necessary to compute the F-statistic that would be used in testing the hypotheses you described in part e.

Short answer

Answer

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

b. 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$E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2$, where$x_1=sizeoforange$and$x_2=typeoforangejuiceextractorused$.

c. 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$E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3$, where$x_1=sizeoforange$and$x_2=typeoforangejuiceextractorused$.

d. Graph

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

f. To do a goodness of fit test is conducted to check the overall goodness of the model where $F-statistic=\frac{R^2/k}{\left(1-R^2\right)/\left[n-\left(k+1\right)\right]}$, Where, $SSE=\sum {\left(y_i-\hat{y}\right)}^2$, $N=no.ofobservations$and $K=no.ofvariablesusedinthe\mathrm{mod}el\left(including{\beta }_0\right)$.

Step-by-step solution

Variable identification

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.

Model equation 

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, $E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2$ where $x_1=sizeoforange$ and $x_2=typeoforangejuiceextractorused$.

Model equation with interaction terms

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 $E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3$, where $x_1=sizeoforange$and $x_2=typeoforangejuiceextractorused$.

Model graph

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.

Hypotheses

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

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

F-statistic

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

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

.