Question: A company produces a standard size can of soda. This means they are labeled as containing 12 fluid ounces. When the volume of cans is labeled, the amount labeled is understood to be the mean amount in all such cans. Suppose that the standard deviation for all the cans this company produces is known to be 0.2306 oz.

A consumer group is suspicious that this company might be trying to save a little money by on average, under filling their cans. They decide to test the company’s claim about the cans containing, on average, 12 ounces. They start by obtaining a random sample of 40 cans of the soda and measuring the contents in each can and then using the mean of this sample as a point estimate for the mean of all the cans.

a) If they decide to accuse the company of cheating their customers provided that the point estimate is less than 12 ounces, then what is the chance they will make a false accusation, assuming that the company actually fills the cans with an average of 12 ounces? In symbols, find P(x bar<12<|µ =12).

b) Does your answer to part (a) require that the population be normally distributed? Explain

The consumer group would not like to take such a risk of making a false accusation, so they decide that they will only accuse the company of under filling their cans if their point estimate is less than 11.9 ounces.

c) If the soda company really does produce an average amount of 12 ounces, what is the chance that the consumer group will make a false accusation under their new criteria? Find P(x bar <11.9|µ=12). Do you think this is a low enough chance of making a false accusation?

Suppose the soda company actually puts only 11.95 ounces in their cans, on average, figuring it will save them money, and the cautious consumer group will be afraid to falsely accuse them of under filling.

d) with this new assumption, what is the chance that the cautious consumer group will get a point estimate that does lead to a correct accusation that company is under filling their cans? Do you think the soda company will get away with it? ( hint: you must find P(x bar<11.9|µ=11.95)

e) What happens to your answer to (d) if the soda company only puts ounces, on average in their cans? (hint: now you must find P(x bar<11.9|µ=11.85) Do you think the soda company will get away with it?

f) What happens to your answer to (d) if the soda company only puts 11.6 ounces, on average in their cans? Do you think the soda company will get away with it?

g) Can a sample of only 40 cans really provide enough evidence to catch a cheating company? Explain