In Applied Life Data Analysis (Wiley, 1982), Wayne Nelson presents the breakdown time of an insulating fluid between electrodes at 34 kV. The times, in minutes, are as follows: 0.25, 0.82, 0.90, 1.21, 2.74, 3.23, 4.15, 4.72, 4.76, 6.40, 7.46, 7.94, 8.33, 12.09, 31.88, 32.47, 33.78, 36.77, and 72.78. Construct a normal probability plot of these data. Does it seem reasonable to assume that breakdown time is normally distributed? Choose the correct answer.

To construct a normal probability plot of the given breakdown times, you would typically follow these steps:

1. Order the data from smallest to largest. 2. Calculate the percentile for each data point using the formula: \[ P = \frac{i - 0.5}{n} \] where \(P\) is the percentile, \(i\) is the rank of the data point, and \(n\) is the total number of observations. 3. Determine the theoretical Z-scores that correspond to the calculated percentiles using the standard normal distribution (Z-distribution). 4. Plot the breakdown times (y-axis) against their corresponding Z-scores (x-axis). 5. Fit a straight line to the plotted points. 6. Analyze the plot.

Looking at the ordered data:

0.25, 0.82, 0.90, 1.21, 2.74, 3.23, 4.15, 4.72, 4.76, 6.40, 7.46, 7.94, 8.33, 12.09, 31.88, 32.47, 33.78, 36.77, and 72.78

Since there are 19 observations, those will be ranked from 1 to 19. Calculate the percentile for each ordered data point using the formula. Then, find the corresponding Z-scores for those percentiles. Once all the data points are plotted on a graph with the Z-scores on the x-axis and the actual breakdown times on the y-axis, the next step is to fit a straight line through the points.

If the points roughly follow a straight line, it's an indication that the data distribution is normal. Any obvious systematic deviations from the straight line might indicate a departure from normality. Particularly, look for points deviating at the ends of the plot, because this could indicate heavy tails or skewness in the distribution.

After constructing the normal probability plot, if the points do not deviate substantially from the straight line that has been fit through them, it would seem reasonable to assume that the breakdown time is normally distributed. Conversely, if there are clear systematic deviations from the line, then the normality assumption may be in question.

Extra: The normal probability plot, also known as a Q-Q (Quantile-Quantile) plot, is a graphical tool to help assess if a dataset is approximately normally distributed. The central limit theorem tells us that many natural phenomena tend to follow a normal distribution, given a large enough sample size.

Understanding normal distribution is essential because it is one of the most fundamental concepts in statistics. It describes a symmetric, bell-shaped curve where most of the observations cluster around the central peak and the probabilities for values further away from the mean taper off equally in both directions.

Normality tests, such as Shapiro-Wilk or Kolmogorov-Smirnov tests, and graphical methods like normal probability plots are commonly used to assess the appropriateness of normality as a modeling assumption for a set of data. In many statistical analyses, particularly parametric tests, an assumption of normality often underlies the application of these methods. If this assumption is not met, then the results of these tests can be misleading.