From page 20 in a well-known applied statistics textbook:
The hypothesis of whether a parameter is positive is directly assessed via its confidence interval. If both ends of the 95% confidence interval exceed zero, then we are at least 95% sure (under the assumptions of the model) that the parameter is positive.
Huh? Who says this sort of thing? Only a complete fool. Or, to be charitable, maybe someone who didn’t carefully think through everything he was writing and let some sloppy thinking slip in.
Just to explain in detail, the above quotation has two errors. First, under the usual assumptions of the classical model, you can’t make any probability statement about the parameter value; all you can do is make an average statement about the coverage of the interval. To say anything about the probability the parameter is positive, you’d need also to assume a prior distribution. And this brings us to the second error: if you assume a uniform prior distribution on the parameter, and the 95% confidence interval excludes zero, then under the normal approximation the posterior probability is at least 97.5%, not 95%, that the parameter is positive. So the above statement is wrong conceptually and also wrong on the technical level. Idiots.
P.S. Elina Numminen sent along the above picture of Maija, who is wistfully imaging the day when textbook writers will get their collective act together and stop spreading misinformation about confidence intervals and p-values.