# Critical Value in Hypothesis Testing: Explained with Examples

The critical value is an important concept in the study of statistical concepts especially in hypothesis testing. It is a point on the test distribution that is employed to determine the acceptance region or the rejection region. The critical value notion explains the range (area) in which a null hypothesis is accepted or rejected. It helps us to apprehend the hypothesis testing, and confidence intervals, and make a useful decision process for the population parameters.

Graphically, the concept of the critical value elaborates on the graph into the region where the null hypothesis is accepted or it is rejected. With the help of this important concept, we can check the statistical significance of test statistics.

In this detailed discussion, we will explore the concept of the critical value. We will discuss the factors influencing critical value, and critical value for one-tailed and two-tailed test statistics. We will address some examples to understand how to compute the critical value in statistical problems.

## What is a Critical Value?

A critical value is a number that elaborates on the rejection zone or region of a hypothesis test. Depending on the kind of hypothesis test you do and the kind of data you are using, different critical values apply.

The critical values in a 95% confidence level hypothesis test known as a two-tailed Z-test are 1.96 and -1.96. If the statistician's results in this test are more than 1.96 or less than -1.96. Next, we choose to accept the alternative hypothesis and reject the null hypothesis.

## Factors Influencing Critical Values:

The critical values depend on the following important factors.

### Test statistic that you are using:

This will vary depending on the sort of data you are using and the kind of research question you have. You will frequently do hypothesis tests in a first-year statistics course using z-statistics, which corresponds to a standard normal distribution, or T-statistics (which corresponds to t-distribution) or chi-squared test statistics, which correspond to a chi-square distribution.

### Selection of significance level:

The person conducting the test will decide this. The likelihood that the null hypothesis will be wrongly rejected when it is true is known as the significance level or alpha level. Selecting a significance level lets you decide how cautious or careful you want to be about preventing this kind of error.

A confidence level may also be used to describe a hypothesis test. Statistical levels and confidence levels are closely connected. One less significant level, or 1-ɑ, is the test's confidence level.

### One-tailed or Two-tailed Test:

The alternative hypothesis will determine whether a hypothesis test is one-tailed or two-tailed. Despite their differences, null and alternative hypotheses are always propositions that contradict one another. The use of a one-tailed upper-tail test is probably appropriate if your alternative hypothesis is confined to positive effects or the right tail of the distribution.

In case your alternative hypothesis (A hypothesis other than the null hypothesis is known as an alternative hypothesis) just focuses on the left tail of the distribution or negative impacts, then a one-tailed lower-tail test will be employed. A one-tailed test contains one critical value and one rejection area, which can be in either the left or right tail of the distribution. The rejection region and critical value of a lower-tail (or left-tailed) test are located in the left tail of the distribution. A right-tailed test, also known as an upper-tail test, has a rejection zone and critical value located in the right tail of the distribution.

In case your alternative hypothesis shows that it deviates from the null hypothesis in any direction, a two-tailed test will be employed. The rejection zone in a two-tailed test is split into two equal sections, one in the distribution's left tail and one in its right tail. There will be a distributional area of size ɑ/2 in each of these rejection regions.

## How to Compute Critical Value (Z & T Critical Value)?

There are various formulas to calculate the critical value, depending on the type of distribution the test statistic is part of. To determine a critical value, consider either the significance level or the confidence interval.

### Z-Critical Value:

When the sample size is more than or equal to thirty and the population standard deviation is known, a z-test is performed on a normal distribution. The important steps to compute this kind of critical value are:

Determine the value of ɑ

Find 1- ɑ for the two-tailed test and 0.5- ɑ for the one-tailed test

The z critical value can be found by looking up the area in the z distribution table. After the computation, the critical value for a left-tailed test must have a negative sign appended to it.

### T-Critical Value:

When the sample size is less than 30 and the population standard deviation (PSD or PSTDEV) is unknown, a t-test is employed. When the population data shows a t-distribution, a t-test is performed. The important steps to calculate t critical value are:

Determine the value of ɑ

Deduct (subtract) 1 from the sample size and it will give us d_{f} (degree of freedom that is one less than the sample size).

Use the one-tailed t-distribution table if the hypothesis test is one-tailed. If not, perform a two-tailed test using the two-tailed t-distribution table.

Alpha value (top row) and matching d_{f} value (left side) in the table should match. To get the t crucial value, locate the junction (intersection) of this row and column.

## Example Section:

#### Example 1:

Determine what will be the z-critical value if ɑ = 0.017 for a right-tailed z-test.

#### SOLUTION:

**Step 1: **The given data is:

ɑ = 0.017

**Step 2: **First of all, subtract the value of ɑ = 0.017 from 0.5 as it is a one-tailed test.

0.5 - ɑ = 0.5 - 0.017 = 0.483

**Step 3: **Now observing the z distribution table:

So,

z = 2.1 + 0.02 = **2.12 Ans**.

#### Example 2:

Let's assume we have 9 sample points of data and we want to perform a one-tailed t-test at ɑ = 0.025. Determine the critical value accordingly.

#### SOLUTION:

**Step 1: **The given data is:

ɑ = 0.025 and sample size (n) = 9

so, d_{f} = 9-1=8

**Step 2: **Now observe the t distribution table:

So,

t (8, 0.025) = **2.306**.

The above calculations is done by using criticalvaluecalculator.com

## Wrap Up:

In this complete discussion, we have explored the concept of the critical value in detail. We have elaborated on the important factors influencing the critical value in hypothesis testing. We have discussed to compute z and t critical values for the one-tailed and the two-tailed tests with examples. Hopefully, by reading this article, you can tackle the problems about the critical values for z and t distributions easily.