A chemical bottling plant produces two types of cleaning fluid concentrates, Alpha and Beta. The volume, XXX centilitres, of liquid in a bottle of Type Alpha follows a normal distribution such that X∼N(10,12)X \sim N(10, 1^2)X∼N(10,12). The volume, YYY centilitres, of liquid in a bottle of Type Beta follows a normal distribution such that Y∼N(15,22)Y \sim N(15, 2^2)Y∼N(15,22). The random variables XXX and YYY are independent.
Find the probability that a randomly selected bottle of Type Alpha contains more liquid than a randomly selected bottle of Type Beta.
A technician selects 5 bottles of Type Alpha and 1 bottle of Type Beta.
Find the probability that the total volume of the 5 bottles of Type Alpha is greater than 3.4 times the volume of the bottle of Type Beta.
The technician defines a linear combination S=k1X+k2YS = k_1 X + k_2 YS=k1X+k2Y, where k1k_1k1 and k2k_2k2 are constants. The technician requires SSS to have a mean of 400 and a minimum variance.
Determine the value of k1k_1k1 and the value of k2k_2k2 that satisfy these requirements.
