Population mean and hypothesis testing

The national average price of a gallon of regular gasoline is 3.71 dollars. Linda would like to assess if the average gas price in her city is significantly higher than the national average. She samples gas prices at 20 local gas stations and records the price of a gallon of regular gasoline. Assume her sample of prices can be considered a random sample (representative of the larger population of all such prices).

What additional condition is required to be able to perform the one-sample t-test?

I know one condition is that the sample is random, but I'm having a hard time figuring out what the second should be. Should the distribution be normal? If so, can I just construct a Q-Q plot to test for this?

Any help would be greatly appreciated.

Should the distribution be normal?

Yes, for the t-statistic to have a t-distribution, the data should be normal; in practice, it won't ever be normal - a price is a strictly non-negative quantity for starters, but generally people hope that approximate normality is sufficient for the purpose of obtaining p-values that are close enough.

If so, can I just construct a Q-Q plot to test for this?

That's a good diagnostic check on the reasonableness of the assumption, but it won't tell you that your data is normal, only that it's not so severely non-normal as to cause undue concern.

Distributionally, your main worry here will tend to be heavy skewness, so you would worry most about seeing a strong curve in a Q-Q plot. Practically, prices tend to be somewhat right-skew, but in this case probably not heavily so. (The t-distribution is also affected by symmetric heavy or light tails but the main effect is to push the significance level up or down a little bit; in general people feel they can tolerate a moderate effect on that).

There are further assumptions - apart from a possible shift in the mean, the distributions the observations are drawn from are the same (basically, with the assumption of normality, this boils down to equality of variance), independence (which the 'random sample' is often considered to take care of, but in some situations won't give it to you; even though it's probably not quite true, it's probably good enough here, but see the comments below that suggest some possible sources of dependence even so; the question is whether these sorts of things are enough to cause a problem).

• Great answer! You might worry that gas stations are not independent even if the sample was random. One example might be the chains. Taking natural log of the price might be useful here to deal with the skew. – Dimitriy V. Masterov Mar 14 '13 at 22:40
• Indeed, the possibility of chains was one thing I worried about - another might be patches of regional variation that leads to a form of dependence. – Glen_b Mar 14 '13 at 22:52