Introduction
Be sure to read through the definitions for this section before trying to make sense out of the following.
The first thing to do when given a claim is to write the claim mathematically (if possible), and decide whether the given claim is the null or alternative hypothesis. If the given claim contains equality, or a statement of no change from the given or accepted condition, then it is the null hypothesis, otherwise, if it represents change, it is the alternative hypothesis.
The following example is not a mathematical example, but may help introduce the concept.
"He's dead, Jim," said Dr. McCoy to Captain Kirk.
Mr. Spock, as the science officer, is put in charge of statistically determining the correctness of Bones' statement and deciding the fate of the crew member (to vaporize or try to revive)
His first step is to arrive at the hypothesis to be tested.
Does the statement represent a change in previous condition?
The correct answer is that there is change. Dead represents a change from the accepted state of alive. The null hypothesis always represents no change. Therefore, the hypotheses are:
States of nature are something that you, as a statistician have no control over. Either it is, or it isn't. This represents the true nature of things.
Possible states of nature (Based on H_{0})
Decisions are something that you have control over. You may make a correct decision or an incorrect decision. It depends on the state of nature as to whether your decision is correct or in error.
Possible decisions (Based on H_{0} ) / conclusions (Based on claim )
There are four possibilities that can occur based on the two possible states of nature and the two decisions which we can make.
Statisticians will never accept the null hypothesis, we will fail to reject. In other words, we'll say that it isn't, or that we don't have enough evidence to say that it isn't, but we'll never say that it is, because someone else might come along with another sample which shows that it isn't and we don't want to be wrong.
State of Nature  
Decision  H_{0} True  H_{0} False 
Reject H_{0}  Patient is alive,
Sufficient evidence of death 
Patient is dead,
Sufficient evidence of death 
Fail to reject H_{0}  Patient is alive,
Insufficient evidence of death 
Patient is dead,
Insufficient evidence of death 
State of Nature  
Decision  H_{0} True  H_{0} False 
Reject H_{0}  Vaporize a live person  Vaporize a dead person 
Fail to reject H_{0}  Try to revive a live person  Try to revive a dead person 
State of Nature  
Decision  H_{0} True  H_{0} False 
Reject H_{0}  Type I Error alpha 
Correct Assessment 
Fail to reject H_{0}  Correct Assessment  Type II Error beta 
Which of the two errors is more serious? Type I or Type II ?
Since Type I is the more serious error (usually), that is the one we concentrate on. We usually pick alpha to be very small (0.05, 0.01). Note: alpha is not a Type I error. Alpha is the probability of committing a Type I error. Likewise beta is the probability of committing a Type II error.
Conclusions are sentence answers which include whether there is enough evidence or not (based on the decision), the level of significance, and whether the original claim is supported or rejected.
Conclusions are based on the original claim, which may be the null or alternative hypotheses. The decisions are always based on the null hypothesis
Original Claim  
Decision 
H_{0} "REJECT" 
H_{1} "SUPPORT" 
Reject H_{0} "SUFFICIENT" 
There is sufficient evidence at the (alpha) level of significance to reject the claim that (restate original claim)  There is sufficient evidence at the (alpha) level of significance to support the claim that (restate original claim) 
Fail to reject H_{0} "INSUFFICIENT" 
There is insufficient evidence at the (alpha) level of significance to reject the claim that (restate original claim)  There is insufficient evidence at the (alpha) level of significance to support the claim that (restate original claim) 
There is one fundamental guideline when performing hypothesis testing.
All hypothesis testing is done under the assumption

Here's how that works
The math used to find these probabilities is explained in a later section. Just try to understand the process and the logic for now.
Now, consider the same problem, except this time, 65 heads were flipped in 100 trials. We find that the probability of that happening, if the coin really is fair, is 0.0027. That's less than 1/2 of one percent of the time. That is unusual. That is so unusual that it is very unlikely to happen by chance alone; therefore, our assumption must be wrong, and we reject our claim that the probability of heads is 1/2, and there is sufficient evidence to reject the claim that the coin is fair.
If the results are too unusual to happen just by chance,

Everybody knows that pi is 3.14, right? Well, Indiana once tried to say it was 3.2 by legislative
action. Here is a little bit lengthier example to test their claim.