Testing the Means of the Grouped Data

The results presented below were calculated using two-sample t-tests on distributions of potassium (K), uranium (eU), and thorium (eTh) measured for the soil samples. The data were grouped based on the classification results of the airborne data from Duval (2005) and the groups were identified as Goliad Formation (Tg), Beaumont Formation (Qb), Lissie Formation (Ql) and/or dune sand (Qds), and recent alluvium (Qal) as interpreted by Duval (2005). One additional group is denoted as "Alt" and corresponds to areas where Duval (2005) indicates that the airborne data are questionable.

The results of the two-sample t-tests are presented in tables (e.g. Table 2) that list the number of samples, mean values, and standard deviations for potassium, uranium, and thorium concentrations in the samples followed by graphs (e.g. Figure 7) that show box-whisker plots (refer to the glossary for more information) as well as histograms and calculated normal probability curves. The t-test results are listed separately in tables (e.g. Table 2.1) below the figures. For each test we are separately comparing the mean values of one of the radioactive elements for two groups. The results provided consider the variances independently and pooled. In either case the information includes the difference in the mean values, the range of differences for a 95 percent confidence interval, the calculated value of t, the degrees of freedom, and the probability value. If the probability values (p) are less than 0.05, the data values are considered to represent different populations at a 95 percent confidence level.

The results of these calculations show that the group interpreted as soils of the Goliad Formation (Tg) is distinct from the soils associated with the Beaumont Formation (see Goliad vs Beaumont), from the Lissie Formation and/or dune sand (see Goliad vs Lissie), and from recent alluvium (see Goliad vs Alluvium) at a 95 percent confidence level. The soil samples associated with the recent alluvium (Qal) form a group that is distinct from those associated with the Beaumont Formation (see Alluvium vs Beaumont) and from the Lissie Formation and/or dune sand (see Alluvium vs Lissie), Although alluvium soils are distinct from the Lissie soils based upon all three radioactive elements, the potassium distributions are not significantly different (refer to Table 5.1). The samples associated with the Beaumont Formation (Qb) are distinct from the Lissie samples but only at a confidence level somewhat greater than 90 percent and less than 95 percent (refer to Table 11.1). Figure 17 is a 3-dimensional view of the estimated means values of K, eU, and eTh for the groups and shows that the sample groups are distinct. Figure 18 plots the mean values with the standard deviations and indicates significant overlap among the sample analyses.

The comparisons of the four samples denoted as Alt indicate that the Alt group is distinct from the Beaumont (refer to Table 7.1) and alluvium (refer to Table 9.1) groups at a 95 percent confidence level and from the (refer to Table 8.1) Lissie group at a confidence level somewhat greater than 90 percent but less than 95 percent. The Alt group is not distinct from the Goliad group (refer to Table 10.1). Because these calculations are made using mean values and standard deviations from only four samples, maximum likelihood estimates were used to determine to which of the other groups the four samples should be assigned. The maximum likelihood calculations indicate that samples ST-142 and ST-143 are members of group Goliad. Sample ST-189 could be a member of group alluvium or Lissie and sample ST-211 is a member of Beaumont.

Two-sample t-tests for Goliad versus alluvium

Table 2. Listing of the mean values and standard deviations of K, eU, and eTh for Goliad versus alluvium.

Figure 7. Graphs comparing the distributions of K, eU, and eTh for Goliad vs alluvium.

 
Table 2.1. Listing of t-test results for the hypothesis that the mean values of the radioelement distributions for groups Goliad and alluvium are equal.

Two-sample t-tests for Goliad versus Beaumont

Table 3. Listing of the mean values and standard deviations of K, eU, and eTh for Goliad versus Beaumont.

Figure 8. Graphs comparing the distributions of K, eU, and eTh for Goliad vs Beaumont.

Table 3.1. Listing of t-test results for the hypothesis that the mean values of the radioelement distributions for groups Goliad and Beaumont are equal.

Two-sample t-tests for Goliad versus Lissie

Table 4. Listing of the mean values and standard deviations of K, eU, and eTh for Goliad versus Lissie.

Figure 9. Graphs comparing the distributions of K, eU, and eTh for Tg vs Lissie.

Table 4.1. Listing of t-test results for the hypothesis that the mean values of the radioelement distributions for groups Goliad and Lissie are equal.

Two-sample t-tests for alluvium versus Lissie

Table 5. Listing of the mean values and standard deviations of K, eU, and eTh for alluvium versus Lissie.

Figure 10. Graphs comparing the distributions of K, eU, and eTh for alluvium  vs Lissie.

Table 5.1. Listing of t-test results for the hypothesis that the mean values of the radioelement distributions for groups Lissie and alluvium are equal.

Two-sample t-tests for alluvium versus Beaumont

Table 6. Listing of the mean values and standard deviations of K, eU, and eTh for alluvium versus Beaumont.

Figure 11. Graphs comparing the distributions of K, eU, and eTh for alluvium  vs Beaumont.

Table 6.1. Listing of t-test results for the hypothesis that the mean values of the radioelement distributions for groups Beaumont and alluvium are equal.

Two-sample t-tests for Alt versus Beaumont

Table 7. Listing of the mean values and standard deviations of K, eU, and eTh for Alt versus Beaumont.

Figure 12. Graphs comparing the distributions of K, eU, and eTh for Alt vs Lissie.

Table 7.1. Listing of t-test results for the hypothesis that the mean values of the radioelement distributions for groups Alt and Lissie are equal.

Two-sample t-tests for Alt versus Lissie

Table 8. Listing of the mean values and standard deviations of K, eU, and eTh for Alt versus Lissie.

Figure 13. Graphs comparing the distributions of K, eU, and eTh for Alt vs Lissie.

Table 8.1. Listing of t-test results for the hypothesis that the mean values of the radioelement distributions for groups Alt and Lissie are equal.

Two-sample t-tests for Alt versus alluvium

Table 9 Listing of the mean values and standard deviations of K, eU, and eTh for Alt versus alluvium.

Figure 14. Graphs comparing the distributions of K, eU, and eTh for Alt vs alluvium.

Table 9.1. Listing of t-test results for the hypothesis that the mean values of the radioelement distributions for groups Alt and alluvium are equal.

Two-sample t-tests for Alt versus Tg

Table 10. Listing of the mean values and standard deviations of K, eU, and eTh for Alt versus Tg.

Figure 15. Graphs comparing the distributions of K, eU, and eTh for Alt vs Tg.

Table 10.1. Listing of t-test results for the hypothesis that the mean values of the radioelement distributions for groups Alt and Tg are equal.

 Two-sample t-tests for Beaumont versus Lissie

Table 11. Listing of the mean values and standard deviations of K, eU, and eTh for alluvium versus Lissie.

Figure 16. Graphs comparing the distributions of K, eU, and eTh for Beaumont vs Lissie.

Table 11.1. Listing of t-test results for the hypothesis that the mean values of the radioelement distributions for groups Beaumontand Lissie are equal.

Figure 17. 3-dimensional graph of estimated mean values for groups of samples.

Figure 18. Graphs of the estimate mean values of K, eU, and eTh for the groups of samples. Horizontal lines represent the standard deviations.