An Introduction to Student's t-Distribution...

We know that we use the central limit theorem and the normal distribution to calculate probabilities where we assume that standard deviation for the population is given. This might be an ideal case as we might not have access to this value in practical situations.

Here is the point where t-distribution steps in. What we do is that we try to predict the population on the basis of sample drawn from it. And use its standard deviation for the actual population.

The equation that governs the process is:

$$T=\frac{\bar{x}-\mu }{s\sqrt{n}}$$

Example:

A chemical engineer clains that the population mean yield of a certain batch process is 500 gm/cu meter of the raw material. To check the claim he samples 25 batches each month. If the computed t-value falls between $\pm t_{0.005}$ then he is satisfied with the claim. What concentration should he draw from a sample that has mean of $\bar{x}=518$ and s=40 gm.

Manual Solution:

Form the table of t-distribution, $\pm t_{0.005}=1.711$ for n = 24 degrees of freedom. The probability is:

$$t=\frac{518-510 }{40\sqrt{25}}=2.18$$

when we look at the value of 2.25 in the body of the t-table under n=24 we found that it has chances of 0.02.

(to be continued...)