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)