statistics and probability

Question

According to a 2009 Reader's Digest article, people throw away about 7% of what they buy at the grocery store. Assume this is the true proportion and you plan to randomly survey 186 grocery shoppers to investigate their behavior. What is the probability that the sample proportion does not exceed 11%?

Answer 1

It is refreshing to see questions posed by students and not just students asking for quiz questions or homework questions to be solved.

The question as it is posed can not be answered for the simple reason that the 7% which is related to an article is not a proportion of responses (counts) but the average of measures of individual consumers waste, either obtained by surveying multiple consumers to get measures of waste (could be anything from none to all) or from external sources (reports on tons of food purchased vs tons of food received by trash centers. In any case it is a average measure of waste.

The 186 grocery shoppers are either being asked how much of what they buy that they throw away (a real measure and considered ratio measure), or they are asked yes or no if they waste food. (or more or less than any specified amount of food). In the first case, without knowledge of the variance of the measures no hypothesis can be tested. In the second case, if we assume count data that will lead to a proportion that can be tested with a binomial distribution (or estimated with a normal distribution), then what we are estimating is going to be based on what the exact question is. There is no binary question that can be asked which can lead to a new estimate for the % of waste as the sample proportion is a proportion of responses (binomial number / nominal measure) not a proportion of waste (real number / ratio measure) .

Another simpler way of proving using proportions from responses to answer a question is the assumption in any binomial distribution is that the probability of an event happening (in this case food being thrown away) must be constant, yet each person who throws away food throws away a different percentage. I'm frugal and eat leftovers and make sure not to let food spoil before being used so I probably do not throw away more than 1%. Some people impulse buy and then don't cook/eat before the food goes bad and they may throw away 25% so the percent thrown away is always different for each person.

Sorry for the technical answer but a "No that don't work" , while easier is much less satisfying.

So - the lesson here is to construct your hypothesis correctly. Develop a method to test (questionnaire or measure) and then answer the hypothesis.

Try again, but look at your lessons before trying to create a question yourself. Good Luck