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# Hypothesis Testing

Statistical hypothesis testing arises in the context in which we observe a random sample of the values of a random variable and from the observations we wish to decide, in a reasoned manner, whether to accept (non-reject) a given hypothesis, called the Null hypothesis, concerning the distribution of the random variable or whether to reject that hypothesis in favor of an Alternative hypothesis also concerning the distribution of the random variable.

To accomplish an hypothesis test, we formulate a test statistic which is a function of the observed values. We partition the possible values the test statistic can take into two sets. If the value of the test statistic falls in the first set, called the critical region, we will reject the Null hypothesis in favor of the Alternative hypothesis. If the value of the test statistic falls in the second set, called the acceptance region, we will reject the Alternative hypothesis in favor of the Null hypothesis.

The discipline of statistics gives us guidelines for how to design a test statistic and how to define the critical region and acceptance region.

## Types of Errors

The discipline of statistics gives us guidelines for how to design a test statistic and how to define the critical region and acceptance region. The guidance is in terms of error.

There are two kinds of errors.
Type 1 error: We may wrongly reject the Null hypothesis when it is true.
Type 2 error: We may wrongly reject the Alternative hypothesis when it is true.

In hypothesis testing, the probability of a type 1 error is fixed by the significance level of the test. If the Null hypothesis is true, it is the probability that the test statistic takes a value in the critical region. Typical values for the fixing of the probability of a type 1 error are 5%, 1%, or .1%. The fixed value is called the significance level of the test.

Of all the possible definitions for a critical region in which the probability that the test statistic will fall into the critical region is fixed ahead of time, which one should be chosen? That answer depends on the Alternative hypothesis. Of all the possible critical regions, each of fixed probability under the Null hypothesis, the one we desire is the one having the highest probability for the test statistic under the Alternative hypothesis. By doing this for a fixed probability of a type 1 error, we minimize the probability for a type 2 error.

## Types of Hypotheses

(1) The experimenter has one key word set.
The Null hypothesis is that this key word set is not encoded; that is, there is no Torah code effect for this key word set. The Null hypothesis says that the ELSs of this key word set, under the given experimental protocol, are in random arrangement as might be expected in a monkey text. The Alternative hypothesis is that the ELSs of this key word set, under the given experimental protocol, are in a more compact arrangement than expected by chance.

(2) The experimenter has one event with multiple key word sets.

Here the key word sets most likely have some common words and they are therefore not independent sets. The Null hypothesis says that each of these key word sets has, under the given experimental protocol, ELSs in a random arrangement as might be expected in a monkey text. The Alternative hypothesis is that one [at least one] of these key word sets, under the given experimental protocol has ELSs in a more compact arrangement than expected by chance.

(3) The experimenter has multiple events each with multiple key word sets.
Here the key word sets associated with the same event most likely have some common words and they are therefore not independent sets. The Null hypothesis says that each of the events has all of its key word sets having ELSs in a random arrangement as might be expected in a monkey text. The Alternative hypothesis is that, under the experimental protocol, all but one of these events have at least one of their key word sets having ELSs in a more compact arrangement than expected by chance.

Technical Discussion: Designing The Test Statistic

Understanding Through Simulated Experiments