What is the t key value and how to calculate it by using the T table?

What is the t key value and how to calculate it by using the T table?

Written by Deepak Bhagat, In Education, Updated On
April 26th, 2024
, 611 Views

In statistics, t critical value is used widely to solve the problems of hypothesis testing. The critical value is that value in hypothesis testing which tells whether to accept or reject the null hypothesis.

The null hypothesis is used in hypothesis testing to ensure that either a statistical relationship or significance exists among two sets of the observed data of the given single observed variable.

In this post, we will learn about the t critical value and how to calculate it by using a formula or by using the t tables.

What is t critical value?

T critical value or simply t value is a value used in statistics to measure the size of the difference relative to the variation in the sample data. It is usually used to determine the change represented in the units of standard error.

The greater magnitude or closer to zero magnitudes of T gives evidence against the null hypothesis that there is a greater significance difference or not. Greater magnitude shows greater evidence that there is a significant difference, while in closer to zero magnitudes, there is no significant difference.

The critical value has two regions: the rejection region, in which the null hypothesis is rejected, and the non-rejection region, in which the null hypothesis is accepted. In simple words, the critical value is the value that cuts the test statistics value on the scale into two regions: the rejection region and the non-rejection region.

While solving the problems of hypothesis testing, if the null hypothesis is rejected then the alternative hypothesis must be accepted. The t critical value can be left-tailed, right-tailed, or two-tailed.

  • Left-tailed test: The critical value from the left to find the area under the density curve is the left-tailed test and is denoted by α.
  • Right-tailed test: The critical value from the right to find the area under the density curve is the right-tailed test and is denoted by α.
  • Two-tailed test: The critical value from the left (α/2) and right (α/2) to find the area under the density curve is a two-tailed test and is denoted by α.
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The degree of freedom must be known to calculate the t critical value using the table. The degree of freedom is a term used to calculate the p-value and t-value for the t-test problems.

How to calculate the t value by using the T table?

If the degree of freedom and significance level are given, the t value or t critical value can be calculated easily using tables. Two types of tables are used to calculate the t value: one for one tail and the other for two-tail values.

T distribution table for one tail

Degree of

freedom

A = 0.10.050.0250.010.0050.0010.0005
ta= 1.2821.6451.9602.3262.5763.0913.291
13.0786.31412.70631.82163.656318.289636.578
21.8862.9204.3036.9659.92522.32831.600
31.6382.3533.1824.5415.84110.21412.924
41.5332.1322.7763.7474.6047.1738.610
51.4762.0152.5713.3654.0325.8946.869
61.4401.9432.4473.1433.7075.2085.959
71.4151.8952.3652.9983.4994.7855.408
81.3971.8602.3062.8963.3554.5015.041
91.3831.8332.2622.8213.2504.2974.781
101.3721.8122.2282.7643.1694.1444.587
111.3631.7962.2012.7183.1064.0254.437
121.3561.7822.1792.6813.0553.9304.318
131.3501.7712.1602.6503.0123.8524.221
141.3451.7612.1452.6242.9773.7874.140
151.3411.7532.1312.6022.9473.7334.073
161.3371.7462.1202.5832.9213.6864.015
171.3331.7402.1102.5672.8983.6463.965
181.3301.7342.1012.5522.8783.6103.922
191.3281.7292.0932.5392.8613.5793.883
201.3251.7252.0862.5282.8453.5523.850
211.3231.7212.0802.5182.8313.5273.819
221.3211.7172.0742.5082.8193.5053.792
231.3191.7142.0692.5002.8073.4853.768
241.3181.7112.0642.4922.7973.4673.745
251.3161.7082.0602.4852.7873.4503.725
10001.2821.6461.9622.3302.5813.0983.300

T distribution table for two tails

Degree of

freedom

A = 0.20.100.050.020.010.0020.001
ta = 1.2821.6451.9602.3262.5763.0913.291
13.0786.31412.70631.82163.656318.289636.578
21.8862.9204.3036.9659.92522.32831.600
31.6382.3533.1824.5415.84110.21412.924
41.5332.1322.7763.7474.6047.1738.610
51.4762.0152.5713.3654.0325.8946.869
61.4401.9432.4473.1433.7075.2085.959
71.4151.8952.3652.9983.4994.7855.408
81.3971.8602.3062.8963.3554.5015.041
91.3831.8332.2622.8213.2504.2974.781
101.3721.8122.2282.7643.1694.1444.587
111.3631.7962.2012.7183.1064.0254.437
121.3561.7822.1792.6813.0553.9304.318
131.3501.7712.1602.6503.0123.8524.221
141.3451.7612.1452.6242.9773.7874.140
151.3411.7532.1312.6022.9473.7334.073
161.3371.7462.1202.5832.9213.6864.015
171.3331.7402.1102.5672.8983.6463.965
181.3301.7342.1012.5522.8783.6103.922
191.3281.7292.0932.5392.8613.5793.883
201.3251.7252.0862.5282.8453.5523.850
301.3101.6972.0422.4572.7503.3853.646
601.2961.6712.0002.3902.6603.2323.460
1201.2891.6581.9802.3582.6173.1603.373
1.2821.6451.9602.3262.5763.0913.291

Example

Determine the t value for one tail and two tails if the degree of freedom is 15 and the significance level is 1%?

Solution

Step 1: Write the given values of degree of freedom and significance level.

Degree of freedom = Df = 15

Significance level = 1% = 1/100 = 0.01

Step 2: Look at the above table of t value for one tail and see the first row of the table for significance level and the first column for the degree of freedom and match the given values and write the corresponding value where both terms intersect.

T critical value for one tail = 2.602

Step 3: Look at the above table of t value for two tails and see the first row of the table for significance level and the first column for the degree of freedom and match the given values and write the corresponding value where both terms intersect.

T critical value for two tails = 2.947

You can also use the t value calculator to find the t value either by one-tail or two-tailed to avoid using larger tables to find the value. You can use this calculator to find the result of your problem according to the tables.

Step 1: You have to put the degree of freedom and significance level in the required box, select one tail or two tail options, and press the “calculate” button.

Step 2: You can get the result below the calculate button in a couple of seconds. This calculator also shows a table up to the given degree of freedom.

Summary

By using the above-mentioned tables, you can easily solve any problem related to this topic. The T value calculations are not tough, you have to find the values from the table. For a higher degree of freedom, you can simply use the above-mentioned calculator.

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