So, the higher the confidence interval the lower the false positive rate, but the false negative rate will increase lowering the recall.
Is it possible to determine which confidence interval is better/more significant? 95% or 80%?
Also why is the lower limit for confidence interval is generally 80%?
Why is it not 70% or 60%?
The language of statistical "significance" is not directly applicable to a confidence interval, though intervals are often related to similar hypothesis tests. If you were to compare a confidence interval to a corresponding hypothesis test you would find that the "confidence level" $1-\alpha$ and the "significance level" $\alpha$ are in fact negatively related --- i.e., a higher confidence level corresponds to a lower significance level. However, saying that a result in a hypothesis test is "more significant" usually means that it has a lower p-value, which is determined by the data, not the chosen significance level. For all these reasons, I strongly recommend that you avoid bringing the terminology of "significance" into discussion of a confidence interval. (It is bad enough in its proper context!)
Just as with significance levels in a hypothesis test, there are no sacrosanct values for the confidence level of a confidence interval. The choice of an appropriate confidence level depends on context, and it always involves a trade-off between desired levels of confidence and accuracy. For a fixed set of data and a fixed procedure for the confidence interval, if you choose a higher confidence level then you will get a wider (less accurate) interval, and vice versa. You can think of a higher confidence level as a more conservative inference, insofar as you are willing to be less accurate in order to be highly confident that the parameter of interest falls within the interval. Contrarily, you can think of a lower confidence interval as a less conservative inference, since you will get better accuracy for the interval but at the cost of having lower confidence that the parameter of interest falls within the interval.
There is no inherent problem with choosing a confidence interval at 60% or 70% confidence level, though obviously that is quite a low level of confidence and so it is a risky inference. Typically people choose higher confidence levels because they want to have a high level of confidence that the unknown value of interest falls within the computed interval. Personally, I tend to be quite conservative in my statistical inferences, so unless there are good contextual reasons to the contrary, if I have a reasonable amount of data I will usually opt for a 99% confidence interval (or perhaps an even higher confidence level).
@Ben gave an excellent answer (+1) I'm just adding to it.
You have to decide which error is worse and by how much. In psychology, where I did most of my work, there's a strong convention for power of 0.8 and alpha of 0.05. This says that a type I error is 4 times worse than a type II error. But that's not always true or even remotely reasonable. Fisher himself said that people should vary their significance levels.
Suppose you come up with a new treatment for a disease that is quickly terminal. Then a type I error means you give a useless treatment to dying people, but a type II error means you fail to save lives that you could have saved.
Now suppose you develop a new treatment for teenage acne. It is much more expensive than current medications and has some nasty side effects. Now a type I error means you cost people money and give them side effects for no reason, but a type II error means a kid has acne for a little bit longer.
Another example of the truth of Sir David Cox's quotation:
There are no routine statistical questions, only questionable statistical routines. He uses statistics the way a drunken man uses a lamp post: More for support than illumination.
The term "significance" should optimally be used referring to test results together with a level indication, such as "the data provide a significant indication against the null hypothesis at level 0.05". Using it with p-values such as "p=0.005 is more significant than p=0.04" is still correct, but somewhat harder to process for non-statisticians. Generally significance always refers to data being incompatible, according to a statistical test, with the data. Otherwise in the interest of clear communication better don't use the word "significant" referring to statistical results.
In particular, the term does not apply to confidence intervals, despite the mathematical relation between confidence levels and significance levels as explained by @Ben. A confidence interval does not regard a single null hypothesis, which is what "significance" is about.
There are no better or worse confidence/significance levels, as there is always a trade-off. The larger the confidence level, the bigger the confidence interval becomes, which means it indicates less precisely where the parameter is, but the smaller the "error probability", i.e., the probability of having the true value not captured in the interval. We want a small interval and a small error probability, so we need to decide how to balance these.
The only thing that can be said is that we want to have a large probability to not miss the true parameter, or, for a test, a small probability to reject a true H0, so confidence levels should be "large" and significance levels "small", but "small" can mean 0.05, 0.01, even 0.001 or 0.1, and values such as 0.019 should only be avoided because they smell of "the author just picked what they needed", so better use widely applied standards, but there isn't anything intrinsically wrong with them.
80% as confidence level is already quite low in my view, it means "1 in 5 cases we go wrong", which, so to say, "can happen all the time", so I wouldn't go that low or even lower, but mathematically also 80% or 65% confidence intervals are not invalid.
