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The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk dens

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asked 2021-05-05

The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with \(\displaystyle\mu={1.5}\) and \(\sigma=0.2g/cm^3\).
(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.
(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than \(0.9g/cm^3\).
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of \(2.0g/cm^3\)? Explain, based on theprobability of this occurring.
(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
(e) What is the moment generating function for X?

Answers (1)

2021-05-07

HERE X IS BULK DENSITY OF PIMA CLAY LOAM
FOLLOW N.D MEAN 1.5 AND S.D \(0.2g/cm^3\)
\(Z =\) (X-MEAN)/S.D IS STANDARD NORMAL VARIATE WITH MEAN '0' AND VARIANCE '1'
PROB.FUNCTION OF N.D
\(\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{\sigma\sqrt{{{2}\pi}}}}}{\exp{{\left[-{\frac{{{1}}}{{{2}}}}{\left({\frac{{{x}-\mu}}{{\sigma}}}\right)}^{{2}}\right]}}}\)
\(\displaystyle={\frac{{{1}}}{{{0.2}\sqrt{{{2}\pi}}}}}{\exp{{\left[-{\frac{{{1}}}{{{2}}}}{\left({\frac{{{x}-{1.5}}}{{{0.2}}}}\right)}^{{2}}\right]}}}\)
image
\(x=1.1\ x=1.5\ x=1.9\)
PROB.OF X LIES BETWEEN 1.1 AND 1.9 IS
\(\displaystyle{P}{\left({1.1}\leq{X}\leq{1.9}\right)}={P}{\left({\frac{{{1.1}-{1.5}}}{{{0.2}}}}\leq{Z}\leq{\frac{{{1.9}-{1.5}}}{{{0.2}}}}\right)}\)
\(\displaystyle={P}{\left(-{2}\leq{Z}\leq{2}\right)}\)
\(\displaystyle={0.9544}\)
b) PROBABILITY LESS THAN 0.9 IS
\(\displaystyle{P}{\left({X}{<}{0.9}\right)}={P}{\left({Z}{<}{\frac{{{0.9}-{1.5}}}{{{0.2}}}}\right)}={P}{\left({Z}{<}-{3}\right)}\)
\(\displaystyle={0.0013}\)
(BY USING STANDARD NORMAL TABLES)
c) BULK DENSITY IN EXCESS OF \(2mg/cm^3\)
HERE \(X>2\)
\(\displaystyle{P}{\left({Z}{>}{\frac{{{2}-{1.5}}}{{{0.2}}}}\right)}={P}{\left({Z}{>}{2.5}\right)}\)
\(\displaystyle={0.0062}\)
YES; BECAUSE THEPROBABILITY OF CLAY DENSITY IS MORE THAN 2 IS 0.0062 (VERYLOW)
SO IT IS SURPRISING THE SAMPLE CONTAINSMORE THAN '2'
d) image
\(\displaystyle{P}{\left({Z}{>}{z}\right)}={0.1}\)
\(\displaystyle{P}{\left({Z}{>}{1.28}\right)}={0.1}\)
(FROM STANDARDNORMAL TABLES)
\(\displaystyle{z}={\frac{{{X}-{1.5}}}{{{0.2}}}}={1.28}\)
\(\displaystyle{X}={1.756}\)
e)
image

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