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 an 85% 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 85% level of confidence, is one region more interesting than the other from a geochemical perspective?

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 positive numbers, we can say that region I is more interesting than region II.    

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

We can not make any conclusions using this confidence interval.

(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

(A) Using TI 84 calculator

enter the first data set in L1 and second data set in L2

press stat then calc then 2var stat

select L1 and L2 as data sets

we get

x1= 747.5, s1 = 170.4, n1 = 12

x2 = 738.9, s2 = 190.2, n2 = 16

(B) using TI 84 calculator

press stat then tests then 2-sampTinterval

enter the data

x1= 747.5, s1 = 170.4, n1 = 12

x2 = 738.9, s2 = 190.2, n2 = 16

Pooled: No

c-level = 0.85

press calculate

we get

(-93.0, 110.2)

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

(D) We have used student's t distribution because we have unknown population standard deviations.

option D