Lognormal Distribution The following are the values of net worth (in thousands of dollars) of recent members of the executive branch of the U.S. government. Test these values for normality, then take the logarithm of each value and test for normality. What do you conclude?

237,592 16,068 15,350 11,712 7304 6037 4483 4367 2658 1361 311

The sample of net worth of recent members of the executive branch of the US government.

The given sample data is arranged in increasing order as:

311, 1361, 2658, 4367, 4483, 6037, 7304, 11712, 15350, 16068, 237592.

In the given data, the sample size is 11. Each value is the proportion of $\frac{1}{11}$of the sample.

So, the cumulative left areas can be expressed in general as $\frac{1}{2n},\frac{3}{2n},\frac{5}{2n}, \text{and} \text{so} \text{on}$.

For the given sample size, , the cumulative left areas, can be expressed as$\frac{1}{22},\frac{3}{22},\frac{5}{22},\frac{7}{22},\frac{9}{22},\frac{11}{22},\frac{13}{22},\frac{15}{22},\frac{17}{22},\frac{19}{22} \text{and} \frac{21}{22}$

.

The cumulative left areas expressed in decimal form are 0.0454, 0.1363, 0.2273, 0.3182, 0.4090, 0.5, 0.5909, 0.6818, 0.7727, 0.8636, and 0.9545.

Refer to the standard normal distribution table, the z-score values corresponding to 0.0454, 0.1363, 0.2273, 0.3182, 0.4090, 0.5, 0.5909, 0.6818, 0.7727, 0.8636, and 0.9545 in a left-tailed is equal to -1.69, -1.10, -0.75, -0.47, -0.23, 0.00, 0.23, 0.47, 0.75, 1.10, 1.69.

Pair thesample values of arm circumferences of females with the corresponding z-score in the form of (x, y) as:

Steps to draw a Normal quantile plot are as follows:

1. Make horizontal axis and vertical axis.
2. Mark the points 32, 33.3, 34.9 up to 45 on the horizontal axis and points -1.3, -1.04, -0.78, up to 1.3.
3. Provide title to horizontal and vertical axis as “X values” and “Z-score” respectively.
4. Mark the co-ordinates and obtain the normality plot as shown below.

From the normal quantile plot, the points do not follow linear pattern. So, the sample of values of net worth of members of the executive branch of the US government does not appear to be from a normally distributed population.

The natural logarithm for the sample data is as:

Now pair thenatural log values of sample with the corresponding z-score in the form of (x, y) as:

Steps to draw a Normal quantile plot are as follows:

1. Make horizontal axis and vertical axis.
2. Mark the points 0, 6, 7 up to 13 on the horizontal axis and points -2.0, -1.5,-1.0 up to 2.
3. Provide title to horizontal and vertical axis as “Log x-values” and “Z-score” respectively.
4. Mark the coordinates on the plot to obtain the normality plot as shown below.

From the normal quantile plot for logarithm values, the points follow a linear and non- symmetric pattern. So, the logarithm values of sample appear to be from a normally distributed population.

The sample values appear to be from a normal distribution after logarithmic transformation. So, the conclusion is that transformation of data leads to normal distribution.