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Rejection region and Acceptance region probability distributions

A rejection region is an area on a graph where the null hypothesis would be rejected, assuming the test results fall in that area. We must remember that the main purpose of statistics is to test theories or the results of experiments. For example, you may have invented a new fertilizer that you think makes plants grow 50% faster. In this regard, to prove that your theory is true, the experiment must be repeatable.

A known fact about plants can be taken. In this example, probably the average growth rate of the plants without the fertilizer. This type of statistical test is called a hypothesis test. The rejection region (also called the critical region) is a part of the testing process. Specifically, it is a probability zone that tells you if your theory, or hypothesis, is probably true.

In this regard, first of all, for a hypothesis test, a researcher collects sample data. From the sample data, the researcher computes a test statistic. If the statistic falls within a specified range of values, the researcher rejects the null hypothesis. The range of values ​​that leads the researcher to reject the null hypothesis is the rejection region.

For example, a researcher might hypothesize that the mean of a population is equal to 10. To test this null hypothesis, he or she might collect a random sample of observations and compute the sample mean. If the sample mean is close to 10 (for example, between 9 and 11), the researcher might decide to accept the hypothesis. In this example, the rejection region would be the range of values ​​that are less than 9 or greater than 11. If the sample mean falls in this range, the researcher would reject the null hypothesis.

Acceptance and rejection regions

All the possible values ​​that a test statistic can assume can be divided into two mutually exclusive groups: one group consisting of the values ​​that appear to be consistent with the null hypothesis, and the other containing the values ​​that are unlikely to occur if Ho is true. The first set is called the acceptance region and the second set of values ​​is known as the rejection region of a test. The value or values ​​that separate the critical region from the acceptance region are called the critical value or values. The critical value, which can be in the same units as the parameter or in the standardized units, must be decided by the experimenter taking into account the degree of confidence that he is willing to have in the null hypothesis.

Rejection regions and probability distributions

Each rejection region can be plotted on a probability distribution. The image above shows a t-distribution with a two-tailed region. It is also possible to have a single queue rejection region.

Two queues versus one queue

The type of test is determined by its null hypothesis statement. For example, if in the example above your statement asks Is the average growth rate of plants greater than 10 cm per day?, this is a one-tailed test, because you are only interested in one direction (greater than 10 cm per day). up to date). You could also have a single rejection region for “less than”. For example, is the growth rate of plants less than 10 cm per day? A two-tailed test, with two regions, would be used when you want to know if there is a difference in both directions (greater than and less than).

Rejection regions and alpha levels

You, as the investigator, choose the alpha level you are willing to accept. For example, if you want to be 95% confident that your results are significant, you would choose an alpha level of 5% (100% – 95%). That 5% level is the rejection region. For a one-tail test, 5% would be in one queue. For a two-tailed test, the rejection region would be two-tailed.

Rejection Regions and P Values

There are two ways to test a hypothesis: with a p value and with a critical value.

p-value method

When a hypothesis test (for example, a z test) is performed, the result of that test will be a p-value. The p-value is a probability value. It’s what tells you whether or not your hypothesis statement is probably true. If the value falls in the rejection region, it means that you have statistically significant results. In this way, you can reject the null hypothesis. If the p-value falls outside the rejection region, it means that your results are not sufficient to rule out the null hypothesis. What is statistically significant? In the plant fertilizer example, a statistically significant result would be one that shows that the fertilizer makes plants grow faster compared to other fertilizers.

In the p-value approach, the probability (p-value) of the numerical value of the test statistic is compared to the specified significance level (α) of the hypothesis test.

The p-value corresponds to the probability of observing sample data at least as extreme as the test statistic actually obtained. Small p-values ​​provide evidence against the null hypothesis. The smaller (closer to 0) the p-value, the stronger the evidence against the null hypothesis.

If the p-value is less than or equal to the specified significance level α, the null hypothesis is rejected. Otherwise, the null hypothesis is not rejected. In other words, if p≤α, H0 is rejected; otherwise, if p>α H0 is not rejected.

In this way, knowing the p-value, any desired level of significance can be evaluated. For example, if the p-value of a hypothesis test is 0.01, the null hypothesis can be rejected at any significance level greater than or equal to 0.01. It is not rejected at any level of significance lower than 0.01. Therefore, the p-value is commonly used to assess the strength of the evidence against the null hypothesis without reference to the level of significance.

Rejection region method with a critical value

The steps are exactly the same as in the previous case. However, instead of calculating a p-value, a critical value is calculated. If the value falls within the region, the null hypothesis is rejected.

Applying the critical value approach determines whether or not the observed test statistic is more extreme than a defined critical value. Therefore, the observed test statistic (calculated from the sample data) is compared to the critical value. It’s kind of a cutoff value. If the test statistic is more extreme than the critical value, the null hypothesis is rejected. If the test statistic is not as extreme as the critical value, the null hypothesis is not rejected. The critical value is calculated based on the given significance level α and the type of probability distribution of the idealized model. The critical value divides the area under the probability distribution curve into the rejection region(s) and the non-rejection region.

In a two-sided test, the null hypothesis is rejected if the test statistic is too small or too large. Therefore, the rejection region of such a test consists of two parts: one on the left and one on the right.

Confidence intervals for rejection

A confidence interval (CI) consists of the possible values ​​that cannot be rejected from a hypothesis test whose H0 is that the value of the observed sample statistic corresponds exactly to the population value it estimates’. IC can be thought of as the “reverse” operation to hypothesis testing.

A typical use of confidence limits is the error bars shown for some sample statistics.

Data analysis consists of comparing the test statistic with that of the null distribution. If the probability that these distributions are equal is small (subject to sampling fluctuations), H0 is rejected. Note that not rejecting H0 does not mean that H0 is true.

The rejection level defines what is unlikely enough and is chosen before testing. The level depends on the specific case, although 5% is a common threshold, and 1% or 10% are also often used. The p-value is the specific probability of the observed value of the test statistic occurring and all others that are at least as unfavorable according to the null distribution.

The rejection region, or the critical region, is the tail (wing) of the pdf that is outside the confidence limits.

Unilateral and bilateral tests

The choice between the two depends on the nature of H0. A one-sided test is used when there is an a priori reason to expect a small or large test statistic (but not both) to violate H0. An example is when the hypothesis is «It rains more in Caracas during the month of September than in Bogotá». A two-sided test is used when very large or small values ​​of the test statistic are unfavorable for H0. For example, a two-sided test is used for H0=”The global mean temperature of Caracas is influenced by sunspots”. The null hypothesis is rejected if the test statistic is greater than $100 (1- α)/2$% or less than $100 (α)/2$%.

Parametric tests and non-parametric tests

Parametric tests make some assumptions about the distribution of the data (usually a theoretical distribution such as Gaussian) and include the Student’s t-test and the likelihood ratio test. Nonparametric (distributionless) tests do not assume a theoretical distribution function, and include range tests such as Wilcoxon-Mann-Whitney, resampling tests, and Monte Carlo integrations.

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