The Canada Public Health Agency (CPHA) is continuously monitoring the volume of mRNA contained in vaccine vials received from one pharmaceutical company. Vials come in two types of color's: purple cap vials which need to be diluted before use and gray cap vials which are diluted before delivery. The messenger RNA (mRNA) content in a purple cap vial of 6 doses is a random variable (X) with mean of 25 mRNA mgs and a population standard deviation of 12 mRNA mg. CPHA usually takes a random sample of 100 purple cap vials for analysis to ensure that doses meet standards.

Questions :

1. The mRNA content in a purple cap vial is considered high when the content is greater than 29 mRNA mgs. A high concentration may generate serious secondary effects when inoculated into human bodies. What is the probability that the resulting sample mean content will be greater than 29 mRNA mgs?

2. Now CPHA would like to implement a similar quality control for the gray cap vials and selects a random sample of 25 gray cap vials from a population with mean content of 25 mRNA mgs, but the standard deviation is now calculated from the sample as 10 mRNA mgs. What is the probability that the sample mean content will be greater than 29 mRNA mgs for this type of vial?

3. CPHA also receives green cap vials of another vaccine produce by another company. The mRNA content of a green cap vial is a random variable with a mean of 26 mRNA mgs and a population standard deviation of 14 mRNA mgs. What is the probability that the mean of a random sample of 100 purple cap vials is lower than the mean of a random sample of 25 green cap vials?