The two examples given below make use of a little bit of math. There's a good reason for that: science uses symbols and numbers to avoid the ambiguity and imprecision of written language. If you're not a math person, it might be rather difficult to follow, but it will be difficult to gain more than a shallow understanding of the scientific method without applying it. Hang on for the ride, and the light bulb that will go on when you see how falsifiability, testing, and predictions all work together will make everything worth it.
This is a statistical application of the scientific method. It makes pretty clear that the goal of the method is not to provide absolute certainty, but rather to provide reasonable certainty. Consider: suppose you wish to determine the distribution of eye color in a given population of ten million individuals. There are two eye colors in the population: brown and blue (ignore the fact that it's an oversimplification; the point is to deal with the process of using the scientific method). What do we do first?
Observe: We see that there are two eye colors in the population. Moreover, we know that brown eyes are dominant and blue eyes are recessive. Therefore, we hypothesize: 75% of the population is brown-eyed. Can we test this hypothesis? Certainly. What we'll do is take a random sample of the population and see if the proportion matches that of the population. Here are the results of our test: 19 out of the 24 members of the randomly-picked people are brown-eyed. This is 79% of the population.
Our sample was random (so there is no sample bias). It's possible to mathematically determine the probability of the sample being a random fluke (I may add another web-page to this effect later on); but for now let us suppose it actually is a fluke. The probability is relatively small, but not overwhelmingly so. For a number, let's say there's a 20% = 0.2 chance the random sample will give a proportion of 75% or more brown-eyed.
We've got some evidence that the population is 75% brown-eyed, but it's not conclusive. We say our hypothesis has passed its first test, but we're only 80% sure that the population is really about 75% brown-eyed. What can we do? We can test it again. So we once more go out and, completely randomly, select from the ten million 28 people, and check their eye color. The results? 20 out of the 28 have brown eyes. This is 71% -- again, very close. If there's a 20% probability that a random sample is a fluke, we've just had two flukes in a row -- the chances of that are 20%*20% = (0.2)(0.2) = 0.04 = 4%. Now we can state with 98% confidence that close to 75% of the population is brown-eyed.
Say we're still not sure. We test another random sample, and we get close to 75% again. The probability of three flukes in a row is (0.2)^3 = 0.008 = 0.8%, so we say we have 99.2% confidence that we're right. It's not 100%, but it's pretty darned close. We haven't proven our hypothesis absolutely, but who would contest that the evidence is conclusive?
Let's wrap this up. What have we seen? The point of the scientific method is to push a hypothesis through test after test so we can gain confidence in its correctness, not so we can prove it outright. Note that it would be quite possible to disprove our hypothesis; all we'd have to do is have several random tests come up with a proportion of brown eyes not close to 75% (indicating that the probability it's actually 75% is quite small). This illustrates the importance of falsification, since it's rather subtle to grasp at first: only by the ability to prove a hypothesis false can we gain any confidence that the evidence points toward it. In this case, we gain confidence by measuring the improbability of successive passes under the assumption the hypothesis is false.
Warning: Math Ahead! If you're not mathematically inclined, try to hang on and understand the concept of what's going on: how the hypothesis is tested against observations.
Sir Isaac Newton, in the mid-1600s, conceived calculus, the three laws of motion, and the law of gravity. With these, he almost single-handedly founded the entire structure of classical mechanics, and pioneered research into the field of optics, too. He stands with Darwin and Einstein as one of the most influential scientists of all time. Let's look at how he tested and supported his law of gravity.
Newton's discovery was, famously, initiated by a falling apple and his realization that the moon is falling toward the Earth. Newton postulated that F = ma -- force is equal to mass times acceleration. This is, of course, the same as a = F/m, or acceleration is equal to force divided by mass. This is his second law. He observed that the force of gravity causes an apple to accelerate toward the Earth; he also observed that the Moon is constantly accelerating toward the Earth (in classical mechanics, any deviation from straight-line, constant-speed motion is an acceleration -- when you swing a foxtail around your head, you're accelerating it toward you even though its speed might remain constant). Based on this, he hypothesized that acceleration is based on gravity. He therefore formulated his universal law of gravitation: every object in the universe attracts every other object with a force proportional to the product of their masses divided by the square of the distance separating them. Mathematically, F ~ Mm/r^2, where M and m are the masses of the bodies and r is the distance separating them. He then hypothesized the existence of a universal gravitational constant (called G) that balances the equation: F = GMm/r^2.
How did Newton test his hypothesis? We'll follow in his footsteps. First, let's note that everything at the surface of the Earth accelerates downward at 9.8 m/s/s. This is a well-established fact. If we can show the hypothesis of universal gravitation predicts an object at the Earth's surface should accelerate at approximately 10 m/s/s toward the center of the Earth, this will be a strong indicator that it is correct: essentially, it will pass the test of every single object ever dropped in the history of science.
Can we go about testing this? Certainly. Suppose we have an object of mass m at the surface of the Earth. We can treat the Earth as a point-mass distance r away (this is why Newton invented calculus: to treat massive objects as though they're simply point-masses at their centers of mass). We know r is about 6000 km. The Earth has mass M = 6e24 kg. G = 6.1e-11 N*m^2/kg^2. So, what does the hypothesis of universal gravitation predict? Well, as pointed out above, a = F/m. Plugging in our hypothesis -- that F = GMm/r^2 -- we have a = GMm/(mr^2) = GM/r^2. We know the values of G, M, and r. A little plugging and chugging gives a = 6e24 kg * 6e-11 N*m^2/kg^2 / (6e6 m)^2 = 3.6e14 / 3.6e13 m/s/s = 1e1 m/s/s = 10 m/s/s. This is precisely what we want!
As in the case above, what are the chances that every single object dropped just happens, by coincidence, to fall at a rate precisely in accord with the predictions of this hypothesis? This is great support -- but we haven't proven the hypothesis completely. Perhaps there's something in the vicinity of the Earth's surface that we haven't accounted for. So can we test the hypothesis against the movement of celestial bodies? Let's test the Moon's motion: if it agrees with Newton's insight, it will be yet more support for the hypothesis of universal gravitation. Supposing (for simplicity's sake) that the Moon's orbit is circular, we know that it must have a centripetal acceleration a = w^2 r, where w = 2pi/T is the frequency and T is the period. We're going to derive a prediction of the period T and see if it is roughly commensurate with the time it actually takes the Moon to go around the Earth -- one month.
From above, we know the acceleration from the Earth's gravitational field is, if the hypothesis is correct, a = GM/r^2, where M is the Earth's mass. So we have a = w^2 r = GM/r^2, and we can derive mathematically (2pi/T)^2 = GM/r^3, or T = 2pi*sqrt(r^3/GM). We know all these values -- r is about 3.8e5 km = 3.8e8 m -- and they predict that the time it takes the Moon to go around the Earth is about 2.4e6 s, or, converting to days and rounding for significant figures, 30 days. This is very strong confirmation of the hypothesis of universal gravitation -- what's the probability that the Moon's motion is also happenstance?
In fact, if you plug in the values for the solar system and the galaxy, the hypothesis of universal gravitation predicts the motion of them all very accurately. The idea that this could all and more be simple coincidence stretches credibility past its breaking point. Though Newton's hypothesis has never -- can never! -- been proven completely, 100%, beyond any doubt true, no reasonable doubt remains that it is a very accurate description of the world in which we live.
To recap: Newton observed the motion of bodies under the influence of gravity, and based on those observations, hypothesized a mathematical relation to describe the motion of bodies with mass. In fact, this hypothesis accurately describes the motion of celestial bodies to almost any degree of accuracy. As in the first case study, no reasonable doubt exists that Newton's hypothesis is not an accurate description of the universe because the probability that we would be incredibly lucky in every test over the past two centuries and have no indication that Newton's hypothesis is wrong is incredibly small.