Thanks for highlighting the issues with false positives and false negatives. Generally referred to as Sensitivity vs Specificity (definitions below).

Generally your specificity decreases the more sensitive your diagnostic / screening test so there's a trade off between missing actual cases and misdiagnosing healthy people.

Best measures for evaluating tests are generally Likelihood ratios (LR) (probability of the person having the disease given the test result) so considers both of the above. LR for a positive test = sensitivity / (1-specificity)

A useful read for anyone interested https://doi.org/10.1136/bmj.315.7107.540 (subscription required)
Greenhalgh, T., 1997. How to read a paper: papers that report diagnostic or screening tests. Bmj, 315(7107), pp.540-543:

"A test is valid if it detects most people with the target disorder (high sensitivity) and excludes most people without the disorder (high specificity), and if a positive test usually indicates that the disorder is present (high positive predictive value)

Sensitivity (True Positive Rate) How good is this test at picking up people who have the condition?
Specificity (True Negative Rate) How good is this test at correctly excluding people without the condition?
Positive predictive value - If a person tests positive, what is the probability that he or she has the condition?
Negative predictive value- If a person tests negative, what is the probability that he or she does not have the condition?"

In general any gentle movement / exercise is better than prolonged bedrest after injury. Recent research and treatment into chronic back pain focuses on RETRAINING (not always strenghthening) the dynamic stability muscles of the deep abdomen and back. (refer to anything recent by C.Richardson P.Hodges J.Hides G.Jull et al). If you are experiencing recurring or chronic back pain the best advice I can give is to SPEAK TO A PHYSIOTHERAPIST and enquire about Transversus abdominus / multifidus retraining. If you want a brief synopsis of recent theory here goes: In normal subjects certain deep stability muscles (not your superficial abdominals or "six pack")automatically contract PRIOR to your body performing an action such as lifting an arm/leg etc. (the proposed neurological pathways and feed-forward mechanisms are still under investigation) In people with chronic back pain, not only have these muscles decreased in size (ie atrophied) but more importantly they no longer contract BEFORE the action or movement occurs. Instead they tend to contract AT THE SAME TIME as the larger superficial muscles which are assisting with moving the limb. Research has shown that with correct training patients can be taught to SELECTIVELY activate/contract stability muscles (rather than the oldfashioned abdominal muscle "bracing") prior to performing tasks which place them at risk. Funnily enough people who practiced this significantly decreased incidence and severity of back pain! The point is - Doing exercise such as situps only reinforces the bad pattern of bracing and late activation of abdominal stabilisers. If you have serious problems GET SERIOUS HELP. See your physio.........