The weights of cans of soup produced by a company are normally distributed with a mean of 15g and a standard deviation of 0.5g.

(a) Find the probability that a randomly selected can of soup will weigh

(i) at least 14.3g and (ii) within 14.3g and 15.3g.

(b) Find the minimum weight of the heaviest 5% of cans of soup produced.

(c) If the soup cans are packed into boxes of 25 cans each, state the distribution of the weigh of a box. Hence find the probability that a randomly selected box weighs less than 370g.

(d) A manager has doubt on the true mean and true standard deviation and so he discards these informations. From the 25 cans in a randomly selected box, he finds the sample mean weight and sample standard deviation are 14.8g and 0.4g respectively.

(i) Construct a 95% confidence interval for the true mean. Based on the confidence interval, is the true mean likely to be 15g?

(ii) If the true mean is 15g, find the probability of observing a sample mean of 14.8g or less. Again, based on this probability, is the true mean likely to be 15g? (Hint: standardize the sample mean of 14.8 before using the t-table.)