Problem 1.2 Statisticians prefer to work with distributions that are symmetrically distributed about a single well-defined peak. When the data are skewed, as is the case for the radiation data (see Figure 1.10), the statisticians Box and Cox have shown that transforming the data by means of the function y = g(x), where g(x) = xλ − 1 λ if λ = 0 g(x) = ln y if λ = 0,

an produce a distribution that is more nearly symmetric for well chosen λ. With respect to the radiation data in Table 1.3 Johnson and Wichern have shown that λ = 0.25 is a good choice. The transformed data is shown in the table below. -1.51 -1.81 -1.39 -1.75 -2.11 -1.65 -1.87 -2.11 -1.87 -1.75 -1.94 -2.50 -2.74 -1.75 -1.75 -1.75 -2.50 -1.75 -2.74 -0.82 -1.75 -2.11 -2.34 -2.11 -1.51 -1.75 -1.51 -1.81 -1.87 -1.39 -1.75 -1.33 -1.70 -1.04 -2.50 -1.33 -1.33 -1.04 -1.04 -0.82 -1.04 -2.11

(a) Make an ordered stem and leaf plot of the data.

(b) Draw the histogram of the data. Use the class width 0.30 and class marks {−2.7, −2.4, −2.1, −1.8, −1.5, −1.2, −0.9}.

(c) Compare the histogram for the transformed data with the histogram for the original data set (Figure 1.10). Is the distribution of the transformed data symmetric or nearly so?