A geologist is exploring the relationship between iron ore concentration and the

Question

A geologist is exploring the relationship between iron ore concentration and the depth of the ore beneath the surface. Sixteen locations were randomly selected at a survey site, and an ore sample was taken at each location. At four randomly selected locations samples were taken at a depth of 1,000 metres, another four randomly selected locations were drilled to 600 metres, and samples at the remaining eight locations were taken at a depth of 400 metres. The sixteen ore samples were then analysed for their concentration of iron ore. The results are available as given below.

a. Perform a parametric test to investigate whether or not the ore concentration is the same across the three depths. Include a complete statistical analysis, including a discussion and full assessment of the assumptions of this particular test. If you conclude that the concentration is not the same, determin where the difference(s) lie. Be sure to clearly explain what analysis you carried out and to include appropriate R output.

b. A parametric test may not be the ideal test here, given the small sample sizes in groups. Carry out an appropriate non-parametric test to investigate whether or not the ore concentration is the same across the three depths. As before, include a complete statistical analysis, including a discussion and full assessment of the assumptions of this particular test, and be sure to include appropriate R output.

c.  It is commonly believed that the true mean iron ore concentration found at 600 metres is given by a weighted average of the true mean iron ore concentrations found at 400 metres and 1,000 metres. In particular, the weight of each of the individual means is determined by how far the drilling depth is from 600 metres. The closer to 600 metres the drilling depth is, the greater the weight corresponding to that particular mean.

(i) Give a reasonable set of hypotheses to test this theory.

(ii) Carry out a test of these hypotheses.

d. (i) What is the relationship between ore concentration and depth?

(ii) Between this model and the model considered in (a) above, which is the better model?

(iii) Provide an estimate for the ore concentration at 1,200 metres. Why might this estimate not be appropriate?

(iv) Provide a reasonable range of values that you might expect to contain the ore concentration for a random sample at 800 metres.

  OreConcentration        Depth  0.354   400  0.28    400  0.745   400  0.387   400  0.413   400  0.261   400  0.508   400  0.643   400  0.574   600  0.745   600  0.367   600  0.45    600  0.689   1000  0.789   1000  0.926   1000  0.647   1000  

Explanation / Answer

a) one-way anova result

since p-value = 0.0227 < 0.05

the ore concentration is not same across the three depths.

for 400 and 600

anova result is

here p-value is 0.43 > 0.05 , hence there is no difference in 400 and 600

similarly

600 and 1000

p-value = 0.068 > 0.05

hence no difference in 600 and 1000

now last 400 and 1000

p-value= 0.009 < 0.05

hence there is difference in 400 and 1000.

b)

b)

Kruskal-Wallis Test: Ore Concentration versus Depth

Kruskal-Wallis Test on Ore Concentration

Depth     N Median Ave Rank      Z
400      8 0.4000       6.1 -2.05
600      4 0.5120       8.4 -0.06
1000      4 0.7390      13.5   2.43
Overall 16               8.5

H = 6.51 DF = 2 P = 0.039
H = 6.52 DF = 2 P = 0.038 (adjusted for ties)

* NOTE * One or more small samples

Anova: Single Factor SUMMARY Groups Count Sum Average Variance 400 8 3.591 0.448875 0.02954327 600 4 2.136 0.534 0.027022 1000 4 3.051 0.76275 0.01539225 ANOVA Source of Variation SS df MS F P-value F crit Between Groups 0.263854125 2 0.131927063 5.13418433 0.022732814 3.805565253 Within Groups 0.334045625 13 0.025695817 Total 0.59789975 15

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