**Overview**

The t-distribution, which is also known as the student’s t-distribution, is the continuous probability distribution. It looks like a normal distribution.

Originally the t-distribution was first published by W.S Gosset in 1908 under the pseudonym of “student”, that’s why it is called student’s t-distribution.

Prof. R.A. Fisher used t-distribution to test the coefficient of regression.

The t-distribution has a bell shaped and symmetric and looks like the normal distribution.

The t -distribution curve is symmetric about the mean zero, uni-modal, bell shaped and asymptotic on both sides of the t -axis.

The tail of t-distribution is fatter than normal distribution and it has higher kurtosis than normal distribution.

**The t-distribution**

The t-distribution is used to test the significance of the mean of small samples, the difference between the means of two small samples and the correlation coefficient. The t -distribution is very useful in tests of hypothesis about one mean, or about equality of two means when σ is unknown.

Suppose are the values of random sample [small sample n<30] taken from a normal population with mean μ and SD (σ) is unknown. If x ̅ be the mean of the sample then a statistic for inference on population mean μ is

Where

**Working rule**

In order to calculate the significance of sample mean at 5% level of significance.

First we calculate the t-statistic

And then compare it to the value of t with (n-1) degrees of freedom at 5% level of significance from the table.

Suppose the tabulated value of t is and if is greater than ‘t’, i.e.>t then we accept the hypothesis.

If , we compare this with the tabulated value of t at 1% level of significance for (n-1) degrees of freedom. Let it is denoted by .

If , the we can say that the value of ‘t’ is significant.

If , we reject the hypothesis.

**Solved example**

**Example: The mean lifetime of a sample of 100 LED lights produced by a company is computed to be 1570 hours with a standard deviation of 120 hours. The company claims that the average life of the LED lights produced by it is 1600 hours. Using the level of significance of 0.05, is the claim acceptable?**

**Solution:**

Here

The t-statistic is-

At 0.05 the level of significance, t = -1.96

-2.5 < -1.96

So we can reject the null hypothesis at 5% level of significance.

Hence the claim is not acceptable.

**Example: A process for making certain ring shaped instruments is under control if the diameters of the instruments have a mean of 0.5000 cm. If a random sample of 10 of these instruments has a mean diameter of 0.5060 cm and SD of 0.0040 cm, is the process under control?**

**Solution:**

Here

The t-statistic is-

Here degrees of freedom = 10 – 1 = 9

At 0.05 the level of significance, t = 3.250

Since- 4.7434 > 3.250, we can say that the process is not under control.

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