Ever wonder how someone in America can be arrested if they really are presumed innocent, why a defendant is found not guilty instead of innocent, or why Americans put up with a justice system which sometimes allows criminals to go free on technicalities? These questions can be understood by examining the similarity of the American justice system to hypothesis testing in statistics and the two types of errors it can produce. (This discussion assumes that the reader has at least been introduced to the normal distribution and its use in hypothesis testing. Also please note that the American justice system is used for convenience. Others are similar in nature such as the British system which inspired the American system)

True, the trial process does not use numerical values while hypothesis testing in statistics does, but both share at least four common elements (other than a lot of jargon that sounds like double talk):

1. The alternative hypothesis - This is the reason a criminal is arrested. Obviously the police don't think the arrested person is innocent or they wouldn't arrest him. In statistics the alternative hypothesis is the hypothesis the researchers wish to evaluate. 
2. The null hypothesis - In the criminal justice system this is the presumption of innocence. In both the judicial system and statistics the null hypothesis indicates that the suspect or treatment didn't do anything. In other words, nothing out of the ordinary happened  The null is the logical opposite of the alternative. For example "not white" is the logical opposite of white. Colors such as red, blue and green as well as black all qualify as "not white".
3. A standard of judgment - In the justice system and statistics there is no possibility of absolute proof and so a standard has to be set for rejecting the null hypothesis. In the justice system the standard is "a reasonable doubt". The null hypothesis has to be rejected beyond a reasonable doubt. In statistics the standard is the maximum acceptable probability that the effect is due to random variability in the data rather than the potential cause being investigated. This standard is often set at 5% which is called the alpha level.
4. A data sample - This is the information evaluated in order to reach a conclusion. As mentioned earlier, the data is usually in numerical form for statistical analysis while it may be in a wide diversity of forms--eye-witness, fiber analysis, fingerprints, DNA analysis, etc.--for the justice system. However in both cases there are standards for how the data must be collected and for what is admissible. Both statistical analysis and the justice system operate on samples of data or in other words partial information because, let's face it, getting the whole truth and nothing but the truth is not possible in the real world.

It only takes one good piece of evidence to send a hypothesis down in flames but an endless amount to prove it correct. If the null is rejected then logically the alternative hypothesis is accepted. This is why both the justice system and statistics concentrate on disproving or rejecting the null hypothesis rather than proving the alternative. It's much easier to do. If a jury rejects the presumption of innocence, the defendant is pronounced guilty. 

Type I errors: Unfortunately, neither the legal system or statistical testing are perfect. A jury sometimes makes an error and an innocent person goes to jail. Statisticians, being highly imaginative, call this a type I error. Civilians call it a travesty. 

In the justice system, failure to reject the presumption of innocence gives the defendant a not guilty verdict. This means only that the standard for rejecting innocence was not met. It does not mean the person really is innocent. It would take an endless amount of evidence to actually prove the null hypothesis of innocence. 

Type II errors: Sometimes, guilty people are set free. Statisticians have given this error the highly imaginative name, type II error. 

Americans find type II errors disturbing but not as horrifying as type I errors. A type I error means that not only has an innocent person been sent to jail but the truly guilty person has gone free. In a sense, a type I error in a trial is twice as bad as a type II error. Needless to say, the American justice system puts a lot of emphasis on avoiding type I errors.  This emphasis on avoiding type I errors, however, is not true in all cases where statistical hypothesis testing is done.

In statistical hypothesis testing used for quality control in manufacturing, the type II error is considered worse than a type I. Here the null hypothesis indicates that the product satisfies the customer's specifications. If the null hypothesis is rejected for a batch of product, it cannot be sold to the customer. Rejecting a good batch by mistake--a type I error--is a very expensive error but not as expensive as failing to reject a bad batch of product--a type II error--and shipping it to a customer. This can result in losing the customer and tarnishing the company's reputation.

In the criminal justice system a measurement of guilt or innocence is packaged in the form of a witness, similar to a data point in statistical analysis. Using this comparison we can talk about sample size in both trials and hypothesis tests. In a hypothesis test a single data point would be a sample size of one and ten data points a sample size of ten. Likewise, in the justice system one witness would be a sample size of one, ten witnesses a sample size ten, and so forth.

Impact on a jury is going to depend on the credibility of the witness as well as the actual testimony. An articulate pillar of the community is going to be more credible to a jury than a stuttering wino, regardless of what he or she says.

The normal distribution shown in figure 1 represents the distribution of testimony for all possible witnesses in a trial for a person who is innocent. Witnesses represented by the left hand tail would be highly credible people who are convinced that the person is innocent. Those represented by the right tail would be highly credible people wrongfully convinced that the person is guilty.

At first glace, the idea that highly credible people could not just be wrong but also adamant about their testimony might seem absurd, but it happens. According to the innocence project, "eyewitness misidentifications contributed to over 75% of the more than 220 wrongful convictions in the United States overturned by post-conviction DNA evidence." Who could possibly be more credible than a rape victim convinced of the identity of her attacker, yet even here mistakes have been documented.

For example, a rape victim mistakenly identified John Jerome White as her attacker even though the actual perpetrator was in the lineup at the time of identification. Thanks to DNA evidence White was eventually exonerated, but only after wrongfully serving 22 years in prison.

If the standard of judgment for evaluating testimony were positioned as shown in figure 2 and only one witness testified, the accused innocent person would be judged guilty (a type I error) if the witnesses testimony was in the red area. Since the normal distribution extends to infinity, type I errors would never be zero even if the standard of judgment were moved to the far right. The only way to prevent all type I errors would be to arrest no one. Unfortunately this would drive the number of unpunished criminals or type II errors through the roof.

Figure 3 shows what happens not only to innocent suspects but also guilty ones when they are arrested and tried for crimes. In this case, the criminals are clearly guilty and face certain punishment if arrested.