A biotechnology company partially finances ongoing research projects at several North American universities. The annual support granted for each project is determined by the progress of the project and has a mean of and a standard deviation of . The company has an annual budget of for supporting research activities. What is the probability that the support granted for projects will not exceed the annual budget? Carry your intermediate computations to at least four decimal places. Report your result to at least three decimal places.

A biotechnology company partially finances ongoing research projects at several North American universities. The annual support granted for each project is determined by the progress of the project and has a mean of $22,115 and a standard deviation of $8,145. The company has an annual budget of $1,000,000 for supporting research activities. What is the probability that the support granted for 44 projects will not exceed the annual budget? Carry your intermediate computations to at least four decimal places. Report your result to at least three decimal places.

We have to calculate the parameters of the distribution of the sum of money for 44 projects.

We apply the rules for the sum of random variables for the mean and standard deviation.

[tex]\mu_s=n\mu=44*22,115=973,060\\\\\sigma_s^2=n\sigma^2\rightarrow \sigma_s=\sqrt{n}\sigma=\sqrt{44}*8,145=6.6332*8,145\\\\ \sigma_s=54,027.8178[/tex]

We can calculate the probabilities of being under the budget as:

[tex]z=(X-\mu_s)/\sigma_s=(1,000,000-973,060)/54,027.8178\\\\z=26,940/54,027.8178=0.4986\\\\P(X<1,000,000)=P(z<0.4986)=0.691[/tex]

It is assumed that the sum of projects money will follow a normal distribution.