Question: Information gathered by the production engineer of FHM, a manufacturer of a liquid clog remover widely used by home owners, revealed that the volume X of liquid contained in the "2,000ml" bottles produced by the company is a normally distributed random variable with mean and standard deviation equal to 2,065ml. and 50ml. respectively.

1. Find the probability that the volume of liquid contained in a randomly selected bottle exceeds 2,000ml. Graph the probability density function of X, clearly and neatly, and show the area that corresponds to this probability.

(b) Find (i) the upper decile and (ii) the lower quartile of the probability distribution of X and explain their meaning. Graph the probability density function of X, neatly and clearly, and show the area that corresponds to these two probabilities.

1. Let M represent the mean liquid soap contained in a random sample of 25 bottles. Write down the probability density function of X and M. Then draw on the same graph the probability density function of X and M. Finally find the probability that M differs from the mean by less than 12ml.
2. Let T be the sum of the volume of liquid contained in a random sample of 16 of these bottles. Find \(E\left( T \right)\) and \(\operatorname{var}\left( T \right)\). Then compute the probability that T exceeds 32,900 ml.

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