Do You Use an Upper or Lower Taile Test When You Want to Know if Something Is Less Than

Choosing whether to perform a one-tailed or a two-tailed hypothesis exam is 1 of the methodology decisions you might demand to make for your statistical analysis. This option tin can take critical implications for the types of effects it can detect, the statistical power of the test, and potential errors.

In this post, y'all'll larn virtually the differences betwixt one-tailed and two-tailed hypothesis tests and their advantages and disadvantages. I include examples of both types of statistical tests. In my next post, I encompass the decision between one and two-tailed tests in more detail.

What Are Tails in a Hypothesis Test?

First, we need to cover some background material to empathize the tails in a exam. Typically, hypothesis tests take all of the sample data and catechumen it to a unmarried value, which is known every bit a exam statistic. You lot're probably already familiar with some test statistics. For instance, t-tests calculate t-values. F-tests, such every bit ANOVA, generate F-values. The chi-square test of independence and some distribution tests produce chi-square values. All of these values are exam statistics. For more information, read my post nigh Test Statistics.

These test statistics follow a sampling distribution. Probability distribution plots brandish the probabilities of obtaining exam statistic values when the null hypothesis is right. On a probability distribution plot, the portion of the shaded surface area nether the curve represents the probability that a value will fall within that range.

The graph beneath displays a sampling distribution for t-values. The ii shaded regions cover the two-tails of the distribution.

Keep in heed that this t-distribution assumes that the null hypothesis is correct for the population. Consequently, the meridian (almost probable value) of the distribution occurs at t=0, which represents the null hypothesis in a t-examination. Typically, the null hypothesis states that at that place is no effect. Equally t-values move further abroad from zero, it represents larger event sizes. When the null hypothesis is true for the population, obtaining samples that exhibit a large credible effect becomes less probable, which is why the probabilities taper off for t-values further from zero.

Related posts: How t-Tests Work and Understanding Probability Distributions

Critical Regions in a Hypothesis Examination

In hypothesis tests, critical regions are ranges of the distributions where the values represent statistically meaning results. Analysts ascertain the size and location of the critical regions by specifying both the significance level (alpha) and whether the test is one-tailed or two-tailed.

Consider the following 2 facts:

• The significance level is the probability of rejecting a aught hypothesis that is correct.
• The sampling distribution for a examination statistic assumes that the cypher hypothesis is correct.

Consequently, to represent the critical regions on the distribution for a test statistic, you only shade the appropriate per centum of the distribution. For the common significance level of 0.05, yous shade 5% of the distribution.

Related posts: Significance Levels and P-values and T-Distribution Tabular array of Critical Values

Two-Tailed Hypothesis Tests

Two-tailed hypothesis tests are besides known as nondirectional and ii-sided tests because you can test for furnishings in both directions. When yous perform a two-tailed test, y'all carve up the significance level percentage between both tails of the distribution. In the example below, I utilize an alpha of 5% and the distribution has two shaded regions of ii.v% (ii * 2.5% = 5%).

When a examination statistic falls in either critical region, your sample data are sufficiently incompatible with the null hypothesis that you can reject it for the population.

In a two-tailed test, the generic null and culling hypotheses are the post-obit:

• Zilch: The effect equals nil.
• Alternative:  The effect does not equal nothing.

The specifics of the hypotheses depend on the blazon of test you perform because you might exist assessing means, proportions, or rates.

Case of a two-tailed 1-sample t-exam

Suppose we perform a two-sided 1-sample t-test where we compare the mean strength (four.1) of parts from a supplier to a target value (5). We apply a two-tailed test because nosotros intendance whether the mean is greater than or less than the target value.

To interpret the results, simply compare the p-value to your significance level. If the p-value is less than the significance level, you know that the test statistic fell into one of the critical regions, but which one? Just look at the estimated result. In the output below, the t-value is negative, so we know that the test statistic savage in the disquisitional region in the left tail of the distribution, indicating the hateful is less than the target value. Now we know this difference is statistically pregnant.

Nosotros tin conclude that the population hateful for office strength is less than the target value. However, the test had the capacity to detect a positive difference equally well. Yous can also assess the confidence interval. With a two-tailed hypothesis test, you'll obtain a two-sided confidence interval. The confidence interval tells the states that the population mean is likely to fall between three.372 and 4.828. This range excludes the target value (v), which is another indicator of significance.

Advantages of two-tailed hypothesis tests

You lot can detect both positive and negative effects. Ii-tailed tests are standard in scientific research where discovering any blazon of effect is normally of involvement to researchers.

1-Tailed Hypothesis Tests

I-tailed hypothesis tests are besides known as directional and one-sided tests because you lot tin test for effects in only one management. When you perform a one-tailed test, the entire significance level percentage goes into the extreme end of ane tail of the distribution.

In the examples below, I use an alpha of five%. Each distribution has one shaded region of 5%. When you perform a one-tailed test, you must determine whether the disquisitional region is in the left tail or the right tail. The test can detect an effect simply in the direction that has the critical region. It has absolutely no capacity to detect an effect in the other direction.

In a one-tailed test, you have two options for the null and culling hypotheses, which corresponds to where you identify the disquisitional region.

Yous can cull either of the post-obit sets of generic hypotheses:

• Null: The effect is less than or equal to zero.
• Culling: The effect is greater than zero.

Or:

• Zip: The effect is greater than or equal to zero.
• Alternative: The outcome is less than zero.

Again, the specifics of the hypotheses depend on the type of exam you lot perform.

Notice how for both possible null hypotheses the tests tin can't distinguish between zero and an outcome in a particular direction. For example, in the instance straight in a higher place, the cypher combines "the result is greater than or equal to naught" into a single category. That exam tin't differentiate between zero and greater than nil.

Example of a 1-tailed 1-sample t-test

Suppose we perform a one-tailed 1-sample t-examination. We'll use a similar scenario as before where we compare the mean strength of parts from a supplier (102) to a target value (100). Imagine that nosotros are because a new parts supplier. Nosotros will use them only if the mean strength of their parts is greater than our target value. At that place is no demand for us to differentiate between whether their parts are every bit strong or less strong than the target value—either way nosotros'd just stick with our current supplier.

Consequently, we'll choose the culling hypothesis that states the hateful difference is greater than zero (Population mean – Target value > 0). The nothing hypothesis states that the difference betwixt the population hateful and target value is less than or equal to zero.

To interpret the results, compare the p-value to your significance level. If the p-value is less than the significance level, you lot know that the exam statistic fell into the critical region. For this study, the statistically pregnant effect supports the notion that the population mean is greater than the target value of 100.

Conviction intervals for a one-tailed test are similarly ane-sided. Yous'll obtain either an upper bound or a lower spring. In this example, nosotros get a lower bound, which indicates that the population mean is likely to be greater than or equal to 100.631. At that place is no upper limit to this range.

A lower-bound matches our goal of determining whether the new parts are stronger than our target value. The fact that the lower bound (100.631) is higher than the target value (100) indicates that these results are statistically meaning.

This test is unable to detect a negative deviation even when the sample mean represents a very negative event.

Advantages and disadvantages of i-tailed hypothesis tests

1-tailed tests take more than statistical power to detect an effect in one direction than a ii-tailed examination with the same blueprint and significance level. One-tailed tests occur most ofttimes for studies where ane of the post-obit is true:

• Effects can exist in only one direction.
• Effects tin can exist in both directions but the researchers but care almost an effect in one management. There is no drawback to failing to detect an effect in the other direction. (Not recommended.)

The disadvantage of one-tailed tests is that they take no statistical power to observe an effect in the other direction.

As role of your pre-study planning process, make up one's mind whether you lot'll use the one- or two-tailed version of a hypothesis test. To learn more than about this planning process, read 5 Steps for Conducting Scientific Studies with Statistical Analyses.

This post explains the differences betwixt one-tailed and two-tailed statistical hypothesis tests. How these forms of hypothesis tests function is clear and based on mathematics. However, at that place is some debate well-nigh when you can use one-tailed tests. My next mail service explores this decision in much more depth and explains the dissimilar schools of thought and my opinion on the matter—When Can I Use One-Tailed Hypothesis Tests.

If you're learning well-nigh hypothesis testing and similar the approach I use in my blog, check out my eBook!

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Source: https://statisticsbyjim.com/hypothesis-testing/one-tailed-two-tailed-hypothesis-tests/

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