a) A company produces a range of snack foods, including cookies. The cookies are produced so that the weight of each cookie is normally distributed with a mean of 25g and a standard deviation of 0.8g. The cookies are then packaged in groups of 12, with an advertised net weight of 295g.

i) If X is the random variable representing the weight of a single cookie, then

identify the type of distribution the random variable X has and write down the value(s) of the parameter(s) of this distribution.

ii) Determine the probability that a randomly selected cookie has a weight of less than 24.5g.

iii) Calculate the proportion of cookies produced with weight greater than 26g.

iv) Calculate the proportion of cookies produced that have a weight within 0.5g of the population mean.

v) Find the weight that 95% of cookies will exceed.

b) The quality control officer for the company is looking at the likelihood of packs of cookies being under the advertised weight. His interest therefore shifts to the mean weight of cookies in a sample of 12, which will be denoted X .

i) State the distribution of the mean of a sample of 12 randomly selected cookies (that is, X ) and give the value(s) of its parameter(s).

ii) Determine the mean weight of cookies in a pack in order for the total contents to be 295g. Hence determine the probability that a randomly selected pack of cookies has a weight that is below the advertised weight.