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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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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?

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
$$\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{\sigma\sqrt{{{2}\pi}}}}}{\exp{{\left[-{\frac{{{1}}}{{{2}}}}{\left({\frac{{{x}-\mu}}{{\sigma}}}\right)}^{{2}}\right]}}}$$
$$\displaystyle={\frac{{{1}}}{{{0.2}\sqrt{{{2}\pi}}}}}{\exp{{\left[-{\frac{{{1}}}{{{2}}}}{\left({\frac{{{x}-{1.5}}}{{{0.2}}}}\right)}^{{2}}\right]}}}$$

$$x=1.1\ x=1.5\ x=1.9$$
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)
$$\displaystyle{P}{\left({Z}{>}{z}\right)}={0.1}$$
$$\displaystyle{P}{\left({Z}{>}{1.28}\right)}={0.1}$$
(FROM STANDARDNORMAL TABLES)
$$\displaystyle{z}={\frac{{{X}-{1.5}}}{{{0.2}}}}={1.28}$$
$$\displaystyle{X}={1.756}$$
e)