Introduction to t-critical value: definition, tables, and examples

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About t-critical Value Definition, Tables, & Examples

In hypothesis testing, the t critical value is frequently used. It allows whether to accept or reject the null hypothesis. If the null hypothesis is rejected, then the alternative hypothesis is accepted. In statistics, hypothesis testing is a way to draw population probability distribution by using the sample data.

While the null hypothesis is an experiment used in hypothesis testing for checking the statistical relationship or significance. In this article, we’ll learn the definition of the t critical value along with a lot of examples.

What is t critical value?

A value that is used to measure the size of the change relative to the variation in the sample statistics data is known as the t critical value. The t critical value is mostly used to measure the difference represented in the units of the standard error.

The evidence of whether the greatest significance difference exists or not in the null hypothesis can be checked by the greater magnitude or closer to zero magnitudes of the t value. If the magnitude of t is greater then there is a greater significant difference.

While if the magnitude of t is closer to zero then there is no significant difference in the null hypothesis. For the acceptance or rejection of the null hypothesis, there are two main regions in critical value. If the null hypothesis is rejected, then the region is called the rejection region.

While if the null hypothesis is accepted, then the region is said to be the non-rejection region. In other words, the critical value has two regions either the rejection region or the non-rejection region that cuts the test statistics value on the scale. 

In the test statistics, the alternative hypothesis must be accepted if the null hypothesis is rejected. The right-tailed, left-tailed, and two-tailed are methods of t critical values. You can use any method to find the t value.

  • The right-tailed test

The right-tailed test is that test in which the values come from the right to measure the area under the density curve. It is denoted by alpha (α).

  • The left tailed test

The left-tailed test is that test in which the values come from the left to measure the area under the density curve. It is denoted by alpha (α).

  • The two-tailed test

The two-tailed test is a test in which half values come from the right (α/2) and the other half values come from the left (α/2). It is denoted by (α).

The degree of freedom (Df) and the significance level (α) must be known to measure the t critical value by using the t table of one-tailed or two-tailed. The degree of freedom (Df) is used to measure the t-value or p-value for t-test problems.

How to measure the t critical value b using the t table of one-tailed or two-tailed?

By using the degree of freedom (Df) and the significance level (α), you can easily find the t critical value from the table. Choose the given degree of freedom from the column and the significance from the rows, then the corresponding value where both the values intersect is said to be the t value.

You can use a critical value calculator to get the result according to the t table of one-tailed or two-tailed. Follow the below steps to use this tool.

Step 1: Select the option i.e., one-tailed or two-tailed.

Step 2: Input the Df and significance level in the boxes.

Step 3: Hit the calculate button below the input box. You will get the result along with the t table up to the given degree of freedom.

T value tables

The one-tailed and two-tailed t tables are given below.

One-tailed t table

Df A = 0.1 0.05 0.025 0.01 0.005 0.001 0.0005
ta= 1.282 1.645 1.960 2.326 2.576 3.091 3.291
1 3.078 6.314 12.706 31.821 63.656 318.289 636.578
2 1.886 2.920 4.303 6.965 9.925 22.328 31.600
3 1.638 2.353 3.182 4.541 5.841 10.214 12.924
4 1.533 2.132 2.776 3.747 4.604 7.173 8.610
5 1.476 2.015 2.571 3.365 4.032 5.894 6.869
6 1.440 1.943 2.447 3.143 3.707 5.208 5.959
7 1.415 1.895 2.365 2.998 3.499 4.785 5.408
8 1.397 1.860 2.306 2.896 3.355 4.501 5.041
9 1.383 1.833 2.262 2.821 3.250 4.297 4.781
10 1.372 1.812 2.228 2.764 3.169 4.144 4.587
11 1.363 1.796 2.201 2.718 3.106 4.025 4.437
12 1.356 1.782 2.179 2.681 3.055 3.930 4.318
13 1.350 1.771 2.160 2.650 3.012 3.852 4.221
14 1.345 1.761 2.145 2.624 2.977 3.787 4.140
15 1.341 1.753 2.131 2.602 2.947 3.733 4.073
16 1.337 1.746 2.120 2.583 2.921 3.686 4.015
17 1.333 1.740 2.110 2.567 2.898 3.646 3.965
18 1.330 1.734 2.101 2.552 2.878 3.610 3.922
19 1.328 1.729 2.093 2.539 2.861 3.579 3.883
20 1.325 1.725 2.086 2.528 2.845 3.552 3.850
21 1.323 1.721 2.080 2.518 2.831 3.527 3.819
22 1.321 1.717 2.074 2.508 2.819 3.505 3.792
23 1.319 1.714 2.069 2.500 2.807 3.485 3.768
24 1.318 1.711 2.064 2.492 2.797 3.467 3.745
25 1.316 1.708 2.060 2.485 2.787 3.450 3.725
1000 1.282 1.646 1.962 2.330 2.581 3.098 3.300

 

Two-tailed t table

Example 1

Find the t critical value by using t tables of one-tailed and the two-tailed if the degree of freedom (Df) is 19 and the significance level (α) 5%.

Solution 

Step 1: Write the given terms in the problem.

The degree of freedom = Df = 19

The significance level = α = 5% = 5/100 = 0.05

Step 2: Use the t table for one-tailed to find the t critical value. First of all, see the column of the degree of freedom and select the given value of Df and then use the first row of the table and match the given significance and take the value where both degrees of freedom and significance level intersect. 

One-tailed t critical value = 1.7401

Step 3: Use the t table for one-tailed to find the t critical value. First of all, see the column of the degree of freedom and select the given value of Df and then use the first row of the table and match the given significance and take the value where both degrees of freedom and significance level intersect.

Two-tailed t critical value = 2.1319

Example 2

Find the t critical value by using t tables of one-tailed and the two-tailed if the degree of freedom (Df) is 103 and the significance level (α) 30%.

Solution 

Step 1: Write the given terms in the problem.

The degree of freedom = Df = 103

The significance level = α = 5% = 30/100 = 3/10 = 0.3

Step 2: Use the t table for one-tailed to find the t critical value. First of all, see the column of the degree of freedom and select the given value of Df and then use the first row of the table and match the given significance and take the value where both degrees of freedom and significance level intersect. 

One-tailed t critical value = 0.5265

Step 3: Use the t table for one-tailed to find the t critical value. First of all, see the column of the degree of freedom and select the given value of Df and then use the first row of the table and match the given significance and take the value where both degrees of freedom and significance level intersect.

Two-tailed t critical value = 1.0422

You can verify the result by using a t value calculator

Summary 

T value is widely used in hypothesis testing. In this article, we have discussed all about the t value along with the tables and solved examples. Now you can solve any problem of t value either by using one-tailed or two-tailed tables.

 

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