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: x_{1}; n_{1} = 12 | |||||
540 | 810 | 790 | 790 | 340 | 800 |
890 | 860 | 820 | 640 | 970 | 720 |
Region II: x_{2}; n_{2} = 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 x_{1}, s_{1}, x_{2}, and s_{2}. (Round your answers to one decimal place.)
x_{1} | = ppm |
s_{1} | = ppm |
x_{2} | = ppm |
s_{2} | = ppm |
(b) Let μ_{1} be the population mean for
x_{1} and let μ_{2} be the
population mean for x_{2}. Find an 80% confidence
interval for μ_{1} − μ_{2}.
(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 80% 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 negative numbers, we can say that region II is more interesting than region I. 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.
(d) Which distribution (standard normal or Student's t)
did you use? Why?
Student's t was used because σ_{1} and σ_{2} are unknown.Student's t was used because σ_{1} and σ_{2} are known. Standard normal was used because σ_{1} and σ_{2} are known.Standard normal was used because σ_{1} and σ_{2} are unknown.
from excel:
a)
x1 =average(Array) =747.5
s1 =stdev(array) =170.4
x2=738.9
s2 =212.1
b)
Point estimate of differnce =x1-x2 = | 8.563 | ||
for 80 % CI & 11 df value of t= | 1.363 | from excel: t.inv(0.9,11) | |
margin of error E=t*std error = | 98.596 | ||
lower bound=mean difference-E = | -90.0 | ||
Upper bound=mean differnce +E = | 107.2 |
Because the interval contains both positive and negative
numbers, we can not say that one region is more interesting than
the other
d)
Student's t was used because σ_{1} and σ_{2} are unknown.
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