A commercial coffee machine dispenses espresso shots and steamed milk for lattes. The volume of an espresso shot, EEE ml, follows the distribution N(30.2,1.22)N(30.2, 1.2^2)N(30.2,1.22). The volume of a portion of steamed milk, MMM ml, follows the distribution N(220.5,4.52)N(220.5, 4.5^2)N(220.5,4.52). A barista prepares a "Large Latte" using a random sample of 3 espresso shots and 2 portions of steamed milk.
Find the probability that the total volume of these 5 components exceeds 540 ml.
A customer orders two portions of steamed milk to be served separately.
Determine the probability that the volumes of these two portions differ by more than 5 ml.
To calibrate the machine, a technician takes a random sample of n+1n+1n+1 espresso shots with volumes E1,E2,E3,…,En+1E_1, E_2, E_3, \dots, E_{n+1}E1,E2,E3,…,En+1. The random variable TTT is defined as
T=nE1−∑r=2n+1Er T = n E_1 - \sum_{r=2}^{n+1} E_r T=nE1−r=2∑n+1ErGiven that P(T>30)=0.0533P(T > 30) = 0.0533P(T>30)=0.0533 to 4 decimal places,
find the value of nnn.
