Chapter 2.7, Problem 226E

a.

To determine

Find the number of variables included in the scatterplot in Figure 2.84(a).

Identify whether each of the variables in Figure 2.84(a) is categorical or quantitative.

Estimate the range for Variable1 and for Variable2.

Answer to Problem 226E

The number of variables included in the scatterplot in Figure 2.84(a) is 2.

Both of the variables are quantitative.

The range for Variable1 is 16.

The range for Variable2 is 90.

Explanation of Solution

From the given scatterplot in Figure 2.84(a), it is clear that there are two variables included, Variable1 in x axis and Variable2 in y axis.

The scales of Variable1 and Variable2 are numerical values. Hence, both of these variables are quantitative.

The minimum and maximum values of the data points observed from the scatterplot for Variable1 are approximately 14 and 29, respectively.

The minimum and maximum values of the data points observed from the scatterplot for Variable2 are approximately 70 and 160, respectively.

The ranges for Variable1 and Variable2 are computed as follows:

$\begin{array}{c}\text{Range}\text{for}\text{Variable1:}\\ Range=29-13\\ =16\\ \text{Range}\text{for}\text{Variable1:}\\ Range=160-70\\ =90\end{array}$

Therefore, the range for Variable1 is 16 and the range for Variable2 is 90.

b.

To determine

Explain whether the association between the variables appears to be positive or negative in Figure 2.84(a).

Answer to Problem 226E

The association between the variables appears to be positive.

Explanation of Solution

In Figure 2.84(a), as the Variable1 increases, Variable2 also increases. This is an indication of positive association.

Therefore, the association between the variables appears to be positive.

c.

To determine

Identify the response variable.

Explain whether the line shows a positive or negative association.

Answer to Problem 226E

The response variable is Variable2.

The regression line shows a positive association.

Explanation of Solution

The variable in the vertical axis represents a response variable and the variable in the horizontal axis represents an explanatory variable.

In Figure 2.84(b), Variable2 is in the vertical axis, whereas Variable1 is in the horizontal axis. Therefore, the response variable is Variable2.

It is also clear from the regression line that the slope of the line is increasing. This indicates that there is a positive association between the variables.

d.

To determine

Identify whether the third variable included is categorical or quantitative.

Find the number of categories if it is a categorical variable.

Find the range if it is a quantitative variable.

Answer to Problem 226E

The third variable included is categorical.

The number of categories is 4.

Explanation of Solution

From Figure 2.85(a), the data points are indicated by different symbols, which are labeled as A, B, C, and D. They are non-numerical values. Thus, it is clear that the Variable3 is a categorical variable.

There are four different labels. Thus, the number of categories is 4.

e.

To determine

Identify whether the association between Variable1 and Variable2 appears to be positive or negative by considering the case in Group A.

Identify whether the association between Variable1 and Variable2 appears to be positive or negative by considering the case in Group B.

Identify whether the association between Variable1 and Variable2 appears to be positive or negative by considering the case in Group C.

Identify whether the association between Variable1 and Variable2 appears to be positive or negative by considering the case in Group D.

Answer to Problem 226E

The association between Variable1 and Variable2 by considering the case in Group A appears to be negative.

The association between Variable1 and Variable2 by considering the case in Group B appears to be negative.

The association between Variable1 and Variable2 by considering the case in Group C appears to be negative.

The association between Variable1 and Variable2 by considering the case in Group D appears to be negative.

Explanation of Solution

From Figure 2.85 (a), it is clear that the data points of the all categories (A, B, C, and D) are in the decreasing order. That is, as Variable1 increases, Variable2 decreases. Thus, the association between Variable1 and Variable2 is negative by considering Group A, Group B, Group C, and Group D.

f.

To determine

Explain whether the regression line for Group A shows a positive or negative association.

Explain whether the regression line for Group B shows a positive or negative association.

Explain whether the regression line for Group C shows a positive or negative association.

Explain whether the regression line for Group D shows a positive or negative association.

Answer to Problem 226E

The regression line for Group A shows a negative association.

The regression line for Group B shows a negative association.

The regression line for Group C shows a negative association.

The regression line for Group D shows a negative association.

Explanation of Solution

In Figure 2.85(b), it is clear from the regression line that the slope of the line is increasing for all the four categories. This indicates the negative association between the variables.

That is, the regression line for Groups A, B, C, and D shows a negative association.

g.

To determine

Explain about the difference in the direction of association between Figure 2.84 and Figure 2.85.

Explanation of Solution

In Figure 2.84, the association between variables is positive, while the association between variables is shown as negative in Figure 2.85.

By including additional information contained in Variable3, the association switches from positive to negative.