Weights of boxes of a particular type of cargo follow some distribution with mea

Question

Weights of boxes of a particular type of cargo follow some distribution with mean 60 pounds and standard deviation 10 pounds (a) If a random sample (i.i.d.) of 140 such boxes are taken, find the probability that the sample average weight will be between 58 pounds and 61 pounds. shipment are i.i.d. Find the 67th percentile of the total shipment weight. can be loaded on to the truck such that the probability is at least 0.96 that the total weight (b) A particular shipment consists of 100 such boxes. Assume the weights of the boxes in the (c) A truck can carry a maximum of 8200 pounds. Find the maximum number of boxes that of the boxes will be at most the maximum 8200 pounds the truck can carry

Explanation / Answer

a)

ere as sample size is greater than 30 we can use normal approximation for which

z score=(Xbar-mean)*sqrt(n)/std deviation

P(58<Xbar<61)=P((58-60)*sqrt(140)/10<Z<(61-60)*sqrt(140)/10)

=P(-2.37<Z<1.18) =0.8810-0.0089 =0.8721

b)std deviation of total weight =10*sqrt(100)=100

and expected total weight =100*60=6000

for 67th pecentile ; critical z =0.44

therefore corresponding weight =mean+Z*std deviation =6000+0.44*100=6044

c)

let number of boxes are n

therefore expected weight =60n and std deviaiton=10*sqrt(n)

also for 96th percentile ; critical z =1.75

therefore P(X<8200)=1.75

(8200-60n)/(10*sqrt(n)) =1.75

8200-60n =17.5*sqrt(n)

60n+17.5*(n)1/2 -8200=0

solving above:

n<=133.3

n=133

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