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?

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

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

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 can not make any conclusions using this confidence interval.

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

a. Standard normal was used because σ₁ and σ₂ are unknown.

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

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

d. Standard normal was used because σ₁ and σ₂ are known.

Answer 1

a)

Following is the output of descriptive statistics

Descriptive statistics
	X1	X2
count	12	16
mean	747.50	738.94
sample standard deviation	170.41	212.15
sample variance	29,038.64	45,006.06
minimum	340	280
maximum	970	993
range	630	713

So we have

B)

The pooled standard deviation:

-------------

So standard error for difference in population mean is

Degree of freedom for t is df=12+16-2=26, so t-critical value using excel function "=TINV(1-0.85,26)" is 1.483. So required confidence interval for   will be

(c)

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

(d)

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