Question

Figure shows three residual plots and a normal probability plot of residuals. For each part, decide whether the graph suggests a violation of one or more of the assumptions for regression inferences. Explain your answers.

Short answer

- The assumption of equal standard deviations is violated in FIGURE (a).

- The assumption of equal standard deviations is violated in FIGURE (b).

- The assumption of equal standard deviations is not violated in FIGURE (c).

- The assumption of linearity is violated in FIGURE (d).

All the points are explained as the graph suggests a violation of one or more of the assumptions for regression inferences.

Step-by-step solution

Given Information

Explanation

The following assumptions are made for regression inferences: Population regression line:

The conditional mean of the response variable $\left(Y\right)$is${\beta }_0+{\beta }_{1}X/30+\text{fix.}$. For each $X$value of the predictor variable.

Standard deviation is the same;

The response variable's standard deviation$\left(Y\right)$is the same as the explanatory variable's standard deviation (X). The standard deviation is denoted by the symbol $sigma.$

Typical populations:

The response variable's distribution is normal.

Observations made independently:

The observations of the response variable are unrelated to one another.

conclusion

- It is clear from FIGURE (a) that the residuals do not fall within the horizontal band. As a result, the linearity assumption may be violated. Furthermore, the residual plot has more variation. Hence. The equality of standard deviations assumption is violated.

- It is clear from FIGURE (b) that the standard deviation is not constant. As a result, the assumption of equal standard deviations is broken.

- It is clear from FIGURE (c) that the residuals are in the horizontal band. As a result, the linearity assumption is not violated. Moreover. The standard deviation is fixed. Hence. The equality of standard deviations assumption is not violated.

- It is clear from FIGURE (d) that the residuals are in the curved pattern. As a result, linearity is assumed.