Question

A soda company advertises that the distribution of ounces in all their cans of soda has a mean of µ = 12 ounces and a standard deviation of σ = 0.2 ounces. Suppose you purchase a case of 30 cans of Sweet Relief (their most popular flavor) and calculate that their average fill is 11.9 ounces. You want to know “What’re the chances!?” You want to use the Central Limit Theorem which you recall tells you that the sampling distribution of the sample mean is normal, has mean µ¯ x = µ, and standard deviation σ¯ x = σ √n (a) (1 point) In order to use CLT we need our sample to be sufficiently large and selected at random. Thirty is a good size but we used a cluster sampling technique. Offer a critique as to why it might not be reasonable to assume our sampling technique estimates simple random sampling for all sodas that the company makes.
(b) (3 points) Assume that it is okay to use the CLT. Describe the shape and determine the center (average) and spread (standard deviation) of the sampling distribution of the sample mean for a sample size of 30.
Shape:
Center:
Spread:
(c) (1 point) Calculate the probability of your case of 30 cans of soda having an average fill of 11.9 ounces or less. Assume that the true average for all cans really is 12 ounces.

Answer 1

Solution

Part (a)

A cluster sample does not truly represent the population fully since only clusters are covered in the sample. Hence, unlike SRS, the probability for each population unit to be included in the sample does not remain the same. Answer 1

Part (b)

Assuming CLT can be applied, Xbar ~ N(12, 0.2/√30) or N(12, 0.0365)

Shape: unimodal symmetric

Center: 12

Spread: 0.0365

Answer 2

Part (c)

Probability of a case of 30 cans of soda having an average fill of 11.9 ounces or less

= P(Xbar ≤ 11.9)

= P[Z ≤ {(11.9 - 12)/0.0365}]

= P(Z ≤ - 2.7383)

= 0.0031 [Using Excel Function: Statistical NORMSDIST] Answer 3

DONE