The weight of an organ in adult has males a bell-shaped distribution with a mean

Question

The weight of an organ in adult has males a bell-shaped distribution with a mean of 300 grams and a standard deviation of 35 grams. Use the empirical rule to determine the following. (a) About 99.7% of organs will be between what weights? (b) What percentage of organs weighs between 230 grams and 370 grams? (c) What percentage of organs weighs less than 230 grams or more than 370 grams? (d) What percentage of organs weighs between 230 grams and 335 grams? (a) and grams (Use ascending order.) (b) % (Type an integer or a decimal.) (c) % (Type an integer or a decimal.) (d) % (Type an integer or decimal rounded to the nearest hundredth as needed.)

Explanation / Answer

here we use stanard normal variate z=(x-mean)/sd

(a) 196.05 and 403.95

here P(|Z|<z)=0.997 , or P(-z<Z<z)=0.997

the corresponding z=2.97 (using ms-excel command=normsinv(0.003/2))

z=-2.97, x=300-2.97*35=196.05

z=2.97, x=300+2.97*35=403.95

(b)for x=230, z=(230-300)/35=-2

for x=370,z=(370-300)/35=2

P(230<X<370)=P(-2<Z<2)=P(Z<2)-P(Z(-2)=0.9773-0.0227=0.9546

required anser is 95.46%

(c)P(X<230)=P(Z<-2)=0.0227

P(X>370)=P(Z>2)=1-P(Z<2)=1-0.9773=0.0227

required probability=P(X<230)+P(X>370)=0.0227+0.0227=0.0454

required answer is 4.54%

(d) for x=335, z=(335-300)/35=1

P(230<X<335)=P(-2<Z<1)=P(Z<1)-P(Z(-2)=0.8413-0.0227=0.8186

required answer is 81.86%

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