4 The elevator in the James Bldg at CBU has a maximum capaci

4. The elevator in the James Bldg at CBU has a maximum capacity of twenty-four hundred pounds. Suppose that ten students get on at the third oor. If the weights of students are normally distributed with a mean of two hundred twenty pounds and a standard deviation of twenty pounds, what is the probability that there will be ten fewer CBU students next semester? SHOW YOUR WORK and use the Z transformation to discuss probability.

Solution

let X be the random variable denoting the weight of student

given that X follows a normal distribution with mean 220pounds and standard deviation 20 pounds

let X1,X2,...,X10 denote the weights of the 10 students who get on the lift at third floor.

so X1,X2,...,X10 is nothing but a random sample of size 10 from X

let S=sum of weights of the 10 students=X1+X2+...+X10

since S is a linear combination of 10 normal variables hence S is also a normal variable with

mean =E[S]=E[X1+E[X2]+...+E[X10]=220*10=2200 pounds

variance=V[S]=V[X1]+V[X2]+....+V[X10]   [since the sample is random hence they are independent and so the covariance terms are zero]

                     =20²*10=4000.

so standard deviation is Sqrt[V[S]]=sqrt(4000)=63.245 pounds

the elevator has a maximum capacity of 2400 pounds.

so the probability that there will be 10 fewer CBU students next semester is

P[the weights of the ten students is greater than 2400 pounds]=P[S>2400]

=P[(S-2200)/63.245>(2400-2200)/63.245]

=P[Z>3.1623]   where Z~N(0,1)

=1-P[Z<3.1623]=1-0.999217=0.000783 [answer]