Question

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

Short answer

The sample of net worth of members of the US government does not follow normality but after taking the logarithm of each sample, the values follow normality.

The conclusion from result after the transformation of the data appear to be from normal distribution.

Step-by-step solution

Given information

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

Arrange the given data in increasing order

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.

Find the Cumulative left areas.

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.

Express the sample values and z-score in (x,y) coordinate .

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

Observed	Z-scores
311	-1.69
1361	-1.10
2658	-0.75
4367	-0.47
4483	-0.23
6037	0.00
7304	0.23
11712	0.47
15350	0.75
16068	1.10
237592	1.69

Make a Normal Quantile Plot  

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.

Conclude from Normal Quantile Plot

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.

Find the natural logarithm values of the given sample

The natural logarithm for the sample data is as:

5.74

7.22

7.89

8.38

8.41

8.71

8.90

9.37

9.64

9.68

12.38

Express the sample values and z-score in (x,y) coordinate

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

Log transformed values	Z-scores
5.74	-1.69
7.22	-1.10
7.89	-0.75
8.38	-0.47
8.41	-0.23
8.71	0.00
8.90	0.23
9.37	0.47
9.64	0.75
9.68	1.10
12.38	1.69

Make a Normal Quantile Plot for logarithm values 

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.

:Conclude from Normal Quantile Plot

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.

Conclude results

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.