The loads hauled by a gravel truck are normally distributed

The loads hauled by a gravel truck are normally distributed with a mean of 18700 pounds and a standard devation of 2300 pounds. Assume all numbers are accurate to a large number of significant digits unless stated otherwise.

(A) What is the probability that the next load will be heavier than 16000 pounds but lighter than 22000 pounds?

(B) What is the bottom quartile for the weight of a load?

(C) Find the probability 6 of the next 15 loads will be lighter than 18000 pounds?

(D) If loads are measured to the nearest 100 pounds, determine the probability that the next load will have a measured wight exceeding 20000 pounds.

Please show step by step on how to do these problems. Thanks

Solution

(A)P(16000<X<22000) = P((16000-18700)/2300<(X-mean)/s<(22000-18700)/2300 )

=P(-11.74<Z<1.43)

=0.9236 (from standard normal table)

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(B) P(X<c)=0.25

--> P(Z<(c-18700)/2300) =0.25

--> (c-18700)/2300= -0.67 (from standard normal table)

So c=18700-0.67*2300=17159

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(C) P(X>18000) = P(Z>(18000-18700)/2300) = P(Z>-0.30)=0.6179 (from standard normla table)

So X follow Binomial distribution with n=15 and p=0.6179

P(X=x)=15Cx*(0.6179^x)*((1-0.6179)^(15-x))

So the probability is

P(X=6)=15C6*(0.6179^6)*((1-0.6179)^(15-6))=0.04836226

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(D) P(X> 20000) = P(Z>(20000-18700)/2300)

=P(Z>0.57) =0.2843 (from standard normal table)