Question

Inorganic phosphorous is a naturally occurring element in all plants and animals, with concentrations increasing progressively up the food chain (fruit < vegetables < cereals < nuts < corpse). Geochemical surveys take soil samples to determine phosphorous content (in ppm, parts per million). A high phosphorous content may or may not indicate an ancient burial site, food storage site, or even a garbage dump. The Hill of Tara is a very important archaeological site in Ireland. It is by legend the seat of Ireland's ancient high kings†. Independent random samples from two regions in Tara gave the following phosphorous measurements (ppm). Assume the population distributions of phosphorous are mound-shaped and symmetric for these two regions.

Region I: x1; n1 = 12
540	810	790	790	340	800
890	860	820	640	970	720

Region II: x2; n2 = 16
750	870	700	810	965	350	895	850
635	955	710	890	520	650	280	993

(a) Use a calculator with mean and standard deviation keys to calculate x₁, s₁, x₂, and s₂. (Round your answers to one decimal place.)

x1	=  ppm
s1	=  ppm
x2	=  ppm
s2	=  ppm

(b) Let μ₁ be the population mean for x₁ and let μ₂ be the population mean for x₂. Find a 99% confidence interval for μ₁ − μ₂. (Round your answers to one decimal place.)

lower limit	ppm
upper limit	ppm

(c) Explain what the confidence interval means in the context of this problem. Does the interval consist of numbers that are all positive? all negative? of different signs? At the 99% level of confidence, is one region more interesting than the other from a geochemical perspective?

Because the interval contains only positive numbers, we can say that region I is more interesting than region II

.We can not make any conclusions using this confidence interval.  

  Because the interval contains both positive and negative numbers, we can not say that one region is more interesting than the other.

Because the interval contains only negative numbers, we can say that region II is more interesting than region I.

(d) Which distribution (standard normal or Student's t) did you use? Why?

Standard normal was used because σ₁ and σ₂ are unknown.

Standard normal was used because σ₁ and σ₂ are known.   

Student's t was used because σ₁ and σ₂ are known

.Student's t was used because σ₁ and σ₂ are unknown.

Answer 1

	Region 1 ( X )		Region 2 ( Y )
	540	43056.25	750	122.3789
	810	3906.25	870	17177.3789
	790	1806.25	700	1516.1289
	790	1806.25	810	5049.8789
	340	166056.25	965	51104.2539
	800	2756.25	350	151272.3789
	890	20306.25	895	24355.5039
	860	12656.25	850	12334.8789
	820	5256.25	635	10803.0039
	640	11556.25	955	46683.0039
	970	49506.25	710	837.3789
	720	756.25	890	22819.8789
			520	47933.6289
			650	7909.8789
			280	210623.6289
			993	64547.75
Total	8970.0	319425	11823	675090.9374

Mean

Standard deviation

Mean

Standard deviation

Confidence interval :-

DF = 25

Lower Limit =
Lower Limit = -193.0746
Upper Limit =
Upper Limit = 210.1996
99% Confidence interval is ( -193.1 , 210.2 )

We can not make any conclusions using this confidence interval.  

  Because the interval contains both positive and negative numbers, we can not say that one region is more interesting than the other.

Student's t was used because σ₁ and σ₂ are unknown.