Question

In a recent year, the average daily circulation of the Wall Street Journal was 2,276,207. Suppose the standard deviation is 70,940. Assume the paper’s daily circulation is normally distributed.

(a) On what percentage of days would circulation pass 1,794,000?

(b) Suppose the paper cannot support the fixed expenses of a full-production setup if the circulation drops below 1,619,000. If the probability of this even occurring is low, the production manager might try to keep the full crew in place and not disrupt operations. How often will this even happen, based on this historical information?

Answer 1

The average daily production of the wall street journal was 2,276,207.

The standard deviation is 70,940.

The paper's daily distribution is normally distributed.

Let, X be the random variable denoting the daily production of papers.

So, X follows the normal distribution with mean 2,276,207 and standard deviation 70,940.

(a) On what percentage of days would circulation pass 1,794,000?

P(X>1,794,000)

=P(X-2,276,207>1,794,000)

=P(X-2,276,207>-482207)

=P(Z>-482207/70940)

Where, Z follows standard normal distribution.

=P(Z>-6.79)

Now, as we know the chance of a standard normal variate lying between -3 and 3 is close to 1, so we can safely say that the chance of the above event is 1.

So, the circulation would pass 1,794,000 on 100% of days, approximately.

(b) Suppose the paper cannot support the fixed expenses of a full-production setup if the circulation drops below 1,619,000. If the probability of this even occurring is low, the production manager might try to keep the full crew in place and not disrupt operations. How often will this even happen, based on this historical information?

ie. to find P(X<1619000)

=P(Z<(1619000-2276207)/70940)

=P(Z<-9.26)

Where, Z is the standard normal variate.

Now, based on the same argument that more than 99% of the standard normal population lies between -3 and 3, so we can say this probability is 0.

So, based on historical info, this will happen approximately 0% of times.