The Shoulder and Knee Clinic knows that their expected monthly revenue from patients depends on their level of advertising. They hire an econometric consultant who reports that their expected monthly revenue, measured in \(\$ 1000\) units, is given by the following equation \(E(R E V E N U E \mid A D V E R T)=100\) +20 ADVERT, where ADVERT is advertising expenditure in \(\$ 1000\) units. The econometric consultant also claims that REVENUE is normally distributed with variance \(\operatorname{var}(R E V E N U E \mid A D V E R T)=900\).

a. Draw a sketch of the relationship between expected REVENUE and ADVERT as ADVERT varies from 0 to 5 .

b. Compute the probability that REVENUE is greater than 110 if \(A D V E R T=2\). Draw a sketch to illustrate your calculation.

c. Compute the probability that REVENUE is greater than 110 if \(A D V E R T=3\).

d. Find the 2.5 and 97.5 percentiles of the distribution of REVENUE when ADVERT \(=2\). What is the probability that REVENUE will fall in this range if \(A D V E R T=2\) ?

e. Compute the level of ADVERT required to ensure that the probability of \(R E V E N U E\) being larger than 110 is 0.95 .