Question

As a packet delivery driver, I load my van with 40 packets each day. The average weight of a packet I carry is 10kg, with a standard deviation of 4kg. If X is the total weight of the packets, what is the probability that this weight is:

a. Exactly 400kg

b. Less than 400kg

c. More than 500kg.

d. Between 450 and 500kg

Answer 1

Let us assume that the weight of each packet follows Normal distribution with the given parameters as mean 10 kg and standard deviation of 4 kg.

Let X be the total weight of 40 packets loaded in a delivery van.

We use following property of Normal distribution to find the distribution of X.

Central limit theorem:

If { Y_i , i=1,2,...,n } are i.i.d. Normal variables with mean= m and Standard deviation = s

then, sum(Y_i) follows Normal with mean= n*m and standard deviation = sqrt(n)*s

Hence, X follows Normal distribution with mean = 40*10 = 400 and standard deviation = sqrt(40)*4 = 25.3

According to the prpoerties of Normal distribution Z=(X-400)/25.3 follows Standard Normal distribution.

Question a) -

We can't use the regular probability tables to calculate the probability of single point.

We use R-studio to find the probability at X=400. Use the following command in R to get the probability.

dnorm(400,mean=400,sd=25.3)

This gives answer, P( X=400 ) = 0.01576

The probability that the weight is exactly 400 kg is 0.01576.

Question b) -

We know that Normal distribution is symmetric around its mean. That is P( X< mean)= ( X>mean) = 0.5

Here the mean of X is 400 kg. Hence, P( X<400 )= 0.5

The probability that the weight is less than 400 kg is 0.5.

Question c) -

Use the following R command to calculate the probability

1-pnorm(3.9526,mean=0,sd=1)

P( X>500 ) = P( (X-400)/25.3 > (500-400)/25.3 ) = P( Z > 3.9526 ) = 0.0000386

Hence,  P( X>500 ) = 0

The probability that the weight is more than 500 kg is almost 0.

Question d) -

P( 450<X<500 ) = P( (450-400)/25.3 <Z< (500-400)/25.3 ) = P( 1.9763 < Z < 3.9526 ) = P(Z<3.9526) - P(Z<1.9763)

= 1 - 0.9772 = 0.0228

The probability that the weight is between 450 and 500 kg is 0.0228.

I hope you find the solution helpful. If you have any doubt then feel free to ask in the comment section.

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Thank you in advance!!!