A diagnostic test is more useful if a negative result correctly identifies absense of disease. We call the proportion of people who don't have a disease and do have a negative test result the specificity of that test.
Let's take another look at our population and test based on its specificity. Set the prior probability of a disease in the population you are studying here: % , and set the specificity of the test here: % . Of course, this therefore also means that (unless the test is 100% specific), that based on the specificity you chose, % in some cases the test will be positive even when there is no disease present - we call these cases false positives).
You can see that when the specificity is high, there will be fewer false positives. (However, you might consider the case where we devise a "test" that doesn't reall measure anything but just indicates "negative" every time it's used: it's specificity will be 100% because it will always provide correct results in people without disease. But what would it's sensitivity be?)