Question

The weights of Delicious Chocolate Bars are approximately normally distributed with mean 54g and standard deviation 14g. They are sold in packs of a 12 bars. One pack of 12 is selected at random. Assume this pack can be regarded as a random sample of 12 chocolate bars. Part A

What is the expected value for the mean weight of chocolate bars in this pack? Give your answer to the nearest whole number in the form x or xx as appropriate.

Answer:

Part B

Consider the average weight of a bar in each pack of 12 chocolate bars. What is the standard deviation of this average weight? (i.e. what is the standard error of the sample mean weights for samples of size 12?) Give your answer to two decimal places in the form x.xx

Answer:

Answer 1

Solution

Let

X = weight (g) of a chocolate bar

T = total weight of 12 chocolate bars in a pack and

Xbar = average weight of a bar in a pack of 12 12 chocolate bars.

Given X ~ N(54, 14²) ................................................................................................................................................. (1)

Back-up Theory

If a random variable X ~ N(µ, σ²), i.e., X has Normal Distribution with mean µ and variance σ², then,

Z = (X - µ)/σ ~ N(0, 1), i.e., Standard Normal Distribution .…...................................….........................…………...…(2)

X bar ~………………………………………………………….................................................................................…….(3)

where X bar is average of a sample of size n from population of X.

If T = Σ(i = 1 to n)xi, then T ~ N[nµ, (nσ)²] ……………....................................................................………....………(4)

Now, to work out the solution,

Part (a)

Vide (4) and (1), expected value for the mean weight of chocolate bars in this pack

= 12 x 54

= 648 g Answer 1

Part (b)

Vide (3) and (1), standard deviation of the average weight of a bar in a pack of 12 bars

= 14/12

= 1.17 g Answer 2

DONE