Calculating a Normal Probability Plot by Hand Consider the following sample data set of n = 5 observations: 14, 11, 17, 15, 13. Complete the following steps to create a normal probability plot.
a. Sort the data from smallest to largest. Use a subscript in parentheses, (i), to represent the ordered data. For example y(1) = 11 is the smallest observation and y(5) = 17 is the largest observed value.
b. For each (i), calculate the (i – 0.5) / n percentile of the standard normal distribution. For example, corresponding to (i) = (1), the (1 – 0.5) / 5 = 10th percentile of the standard normal distribution is – 1.28, since P(Z … – 1.28) = 0.10 when Z ~ N(0, 1). For (i) = (3), the (3 – 0.5) / 5 = 50th per-centile (i. e., the median) of the standard normal distribution is 0. Repeat this process for the other ordered values, y(2), y(4), and y(5).
c. Make a normal probability plot by creating a scatterplot with the percentiles of the observed data along the x-axis and the percentiles of the standard normal distribution along the y-axis. If the data fall along a straight line, then the data are consistent with the hypothesis that they are a random sample from a population that is normally distributed. The data in this question are a little “heavier” toward the tails (the normal distribution has more observations in the center and fewer observations toward the tails than does this data set), so the probability plot has an S-shaped curve. With only five data points, the shape is not as clear as it would be for a data set with a larger sample size from a “heavy- tailed” population.
d. If you standardized the data (subtracted the sample mean and then divided by the sample standard deviation), would you expect the shape of the normal probability plot to change?
e. Does the shape of the normal probability plot change if you multiply each observation in the sample data set by 5?
f. Does the shape of the normal probability plot change if you divide each observation in the sample data set by 3?