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take a look at this question:

The sample from Population A has a mean of 35 and a standard deviation of 1. The sample from Population B has a mean of 45 and a standard deviation of 15. Which of the following are certain?

and there is this option:

The average age of Population A is lower than the average age of Population B.

the grader tells that this answer is not correct.

But I can't see why. if population A has SD of just 1 and it mean is 35, how come it can ever have an average population lower than B?

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What is the definition of average age? – gammatester Apr 7 '14 at 11:04
@gammatester I don't have it.. the first part of the question reads as: "You are taking samples of the ages of two populations, A and B. Population A is all the residents of San Francisco, while Population B is all the residents of Los Angeles." – Draconar Apr 7 '14 at 11:09
You could not be certain of this option as you can only make statements about probability like there is a 95% chance that the average age of A is lower than B. To be certain you would have to survey the whole population without error. – user121049 Apr 7 '14 at 11:18

1 Answer 1

up vote 1 down vote accepted

They key word in the question is sample. What this means is that from each of the two populations $A$ and $B$, a subset of individuals were selected from each, and the mean and standard deviations for each population were calculated for these people only.

Therefore, it may not necessarily be the case that the samples are truly representative of the populations from which they were drawn. It is possible that there was sampling bias or that mere sampling variation resulted in a sample mean and standard deviation that does not reflect the true population mean and standard deviation.

For example, suppose I am interested in the distribution of ages of residents of New York state, compared to that of Florida, as of January 1, 2014. On this day, the population mean and standard deviation are fixed, but unknown to us. It is impractical to ask every single person in each state, so typically, a statistician would take a random sample and obtain an estimate of these true population parameters. For example, I might ask choose 100 people from the phone books of each state; or I might ask 100 people on the street.

Now, some sampling methods are better than others, in the sense that they aren't as likely to result in bias. Clearly, if I went to nightclubs in NY and asked for the patrons' ages, and then I went to Florida retirement homes and asked for their ages, I would introduce a great deal of sampling bias in each sample I collected--the point is that the samples I took may not faithfully represent the true distribution of age of residents for each state. But even if I did somehow collect truly random samples, there is still a possibility that the sample means and standard deviations I calculate will not be close to the population means and standard deviations, simply because there is randomness inherent in the sampling process. This notion is analogous to the idea that, if I toss a fair coin 10 times, I am not guaranteed to always obtain exactly 5 heads and 5 tails--the randomness of each trial means I could get 7 heads and 3 tails, or 4 heads and 6 tails, or even 10 heads and 0 tails, simply by chance. The same is true of sampling variation.

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