Question

What diagnostic plot can you use to determine whether the data satisfy the normality assumption? What should the plot look like for normal residuals?

Short answer

Answer: A Normal Probability Plot, also known as a Q-Q Plot, is a graphical method used to determine if a dataset follows a normal distribution or not. It is created by plotting the observed quantiles of the data against the quantiles from the expected normal distribution. To determine if the dataset satisfies the normality assumption, we focus on the residuals, which represent the differences between the observed values and the expected normal values. If the plotted points in the Normal Probability Plot approximately form a straight line, then the residuals follow a normal distribution and the data satisfies the normality assumption. However, if the points significantly deviate from a straight line, the normality assumption may not be satisfied.

Step-by-step solution

Understand the Normal Probability Plot

A Normal Probability Plot, also known as a Q-Q Plot, is a graphical method used to determine if a dataset follows a normal distribution or not. In this plot, the observed quantiles of the data are plotted against the quantiles from the expected normal distribution.

Understand the Residuals

The residuals are the differences between the observed values and the expected values from a statistical model. In this case, we are interested in the normal residuals, which are the differences between the observed values and the expected normal values. When the residuals are normally distributed, it indicates that the data satisfies the normality assumption.

Create the Normal Probability Plot

To create the Normal Probability Plot, follow these steps: 1. Arrange the observed data in ascending order. 2. Calculate the percentile for each data point using the formula: \(Percentile = \frac{rank}{(n+1)}\), where "rank" refers to the position of the data point in the dataset and "n" is the total number of data points. 3. Calculate the z-score for each percentile using the inverse cumulative distribution function for a standard normal distribution. 4. Plot the observed data (on the x-axis) against the corresponding z-scores (on the y-axis).

Interpret the Normal Probability Plot

To determine if the residuals follow a normal distribution, look at the plotted points in the Normal Probability Plot. If these points approximately form a straight line, then the residuals follow a normal distribution and the data satisfies the normality assumption. If the points significantly deviate from a straight line, the normality assumption may not be satisfied.