A 95% confidence interval is usually sufficient for sample estimates of most opinion polls or population parameters. If the level of confidence increases, then the confidence interval becomes less precise (widens). Choosing between 95% and 99% is a trade-off between reducing risk (and increasing confidence) versus the width of the interval.
For example, in the chapter exercises, Question #3 is about opinions on global warming. The data show that 589 out of the 1,511 surveyed felt global warming is a very serious problem. When calculating the proportion of respondents, the 95% confidence interval is between 0.365 to 0.415, and the 99% between 0.356 to 0.424. In this particular case, the intervals have a very small difference. Is there a good rationale for choosing 95% versus 99%? I think the discussion context might matter here more than the numbers. A news article seems okay for reporting the 95% confidence interval. A high-level administrator trying to make an evidence based decision about federal environmental science investments might want to use the 99% (or even 99.9% for a large budget worth billions of dollars).
The other two factors involved in choosing a confidence interval might be sample size and sample standard deviation. A larger sample size increases the precision of the confidence interval (narrows), but also increases time and cost for the survey. A higher sample standard deviation decreases the precision of the confidence interval (widens). Both of these might affect choosing between the result of a 95% confidence interval vs 99% as well as the reporting venue.
For a 99.9% confidence interval, this means that the researcher really wants to be as close to 100% as possible (since 100% doesn’t actually seem possible in estimation). Health decisions might fit within this category, such as estimating the spread of deadly diseases in the country, as well as ones involving billions of dollars. If there’s a rule of thumb for using a certain percent, I haven’t come across it and probably won’t because it’s too dependent on context.
For an internet resource, I looked at Stattrek: http://stattrek.com/estimation/confidence-interval.aspx. I’ll admit what drew me in is the similarity to Star Trek. Unfortunately, the website doesn’t use science fiction references or problem examples. The estimation section includes more on margin of error than our textbook, such as finding the critical value and expressing it as a t score (or z score). The section on confidence intervals is relatively short, simple, and compatible with our textbook. The estimation problems also go further than our text with regression slopes and calculating differences (between proportions, means, and matched data pairs).